(Whether there are exactly two standard positions, or actually more in some cases, doesn't affect my point.)
Popper's epistemology is the most innovative epistemology of note. By that I mean it changes more from prior epistemology than any of its rivals do. It's the most different. That makes it harder to understand.
(Also, to be clear, Popper personally is not important. Like all philosophers, different people have read his books and interpreted him to be saying a variety of different things. I am interested here only in what I regard as the correct, best interpretation. This includes refinements by David Deutsch and myself.)
What commonly happens is Popper (or a Popperian, or a person advocating a Popperian idea, whatever) says a particular epistemology idea is mistaken and tries to explain why. Then people usually interpret Popper as being on the other side of the dichotomy from them, because he's disagreeing with them. "If he says I'm wrong, he must be on the opposing side from me!" That's an easy conclusion to reach when you don't fully understand the point being made. But actually Popper is taking neither of the standard sides.
It's hard conceiving of a new way of looking at an issue. That's harder than understanding that someone has an opposing position which you've heard before and have arguments about. The standard opponent is within your framework, which is easier to deal with.
Look at it another way. For many issues, there are two sides which disagree but also have some points of agreement. For example, they agree on what the right question or dichotomy is, but give opposing answers to it. When popper says that not only is their answer wrong, but also their question is wrong, Popper is disagreeing with them more than their opponents do! So he could be misunderstood as an even more disagreeable version of their opponents, even though he isn't.
This is relevant to Objectivism because Objectivists have misunderstood Popper, and their criticisms of Popper rely on misunderstanding his positions. There aren't any Objectivist refutations of the Popperian ideas I'm advocating. (Nor are there Objectivist answers to Popper's actual criticisms of some Objectivist positions, like induction).
Popperian epistemology does not contradict all of Objectivist epistemology. There are many points in common, such as valuing clarity and accepting the possibility of humans attaining objective knowledge. But there are some major points of disagreement such as induction and self-evident axioms. Objectivists have the opportunity to learn something, and should be happy about that (just as, for example, Popperians could and should learn a lot from Objectivist morality and politics).
Let's look at some example issues where there is a false dichotomy which Popper rejects: certainty and proof, induction, justification, support.
Take certainty or proof: there is a false dichotomy between having certainty and not having knowledge. There is an assumption, shared by both sides, that certainty is a requirement of knowledge. Popperian epistemology rejects that package deal, and offers a new way: a non-authoritarian, fallibilist way to gain objective knowledge.
Take induction: the two main positions both center around the problem of induction. One position is that we can solve the problem of induction (some claim they already did solve it, some expect it to be solved any decade now). Another position is that the lack of solution to the problem of induction presents a big problem for epistemology. The popperian position is that it's the wrong problem, the wrong question. Popper instead raised a different better question and solved it.
Take justification: there is a false dichotomy between "yes we can justify our beliefs/ideas/knowledge" and "no we can't, justification fails due to regress [and several other arguments], therefore knowledge is impossible". The Popperian view is that both of these positions are wrong. They both agree on an incorrect concept of what justification is and why we need it. They package justification together with knowledge.
Take support: consider the idea that we can support our beliefs with evidence and arguments. Some people say we can't, therefore our beliefs are irrational. Some people say we can, and it makes our beliefs rational. Both sides have accepted that we need to support our beliefs with evidence and argument for them to be rational. Popper disagrees with both standard sides. He says we don't have to support our beliefs with evidence and argument for them to be rational; that isn't actually how you get rational knowledge; but there is a different way of getting rational knowledge.
There is a package deal combining rationality and support. And it creates a false dichotomy where either you have both rationality and support, or neither.
Popperian epistemology is a complex subject requiring study to understand well. I cannot cover it all here. I'm going to talk about one example in more detail to give you a sample.
Do we have to support our beliefs with evidence and arguments for them to be rational? Pretty much everyone agrees the answer is "yes". That includes both people who think we can do this and thereby get rational knowledge, and also people who think that our inability to do this prevents us from getting rational knowledge (skeptics).
The Popperian view is that rationality is not about support. It is achieved by a different method. Rational ideas are ideas which are open to criticism. If there's no way to improve an idea, it's stuck, it's bad, it's irrational. If it's open to improvement via criticism – if it's open to reform, refinement, error correction – then it is rational.
Whether ideas are open to error correction does not depend on how much support they have. That is not the issue. (And actually, sometimes when people say, "I've proved my case with all this supporting evidence," it can indicate they are not open to criticism.)
Think, for a moment, about what we want to accomplish in epistemology. For example: we want to sort out good ideas from bad ideas. We want to improve our ideas. We want to get knowledge – ideas that are connected to reality and effective in reality.
Trying to support ideas was a false goal. It's not really what we wanted. It was a way of getting something else. It had indirect value. It's important to identify this gap and separate the concepts. We can reject support but still find a different method to get the useful stuff support was intended to achieve.
Supporting ideas is meant to sort out good ideas from bad ideas. The ones with more support are good. This method does not work. One unsolved problem with it is to define exactly when, why and how much any given idea supports any other ideas. A second problem is whether a less supported idea could be the best one. If it can, what does it really matter that it's less supported?
However, a different method of sorting out good ideas does work: criticism. Ideas which are not refuted by criticism are sorted out from those which are refuted by criticism. (These critical classifications are always open to revision in the future as we learn more.)