I believe this is one of the most important question in philosophy. It questions whether philosophy is worth doing.
The solution to the problem lies in seeing that Kant and Popper deny the possibility of certain knowledge, as does their tradition of thought. Thus the question of truth for them is confusing. This denial of the possibility of knowledge suffuses the thinking of our local tradition of philosophy and we see it in some of the comments here.
They speak of truth in terms of correspondence or coherence, or as requiring 'justified true belief', which must be the most muddled idea in philosophy. They argue quite rightly that we cannot claim the truth of a philosophical or scientific theory since a scientific or metaphysical theory is just a likely story, as one ancient Greek notes. It may be the best theory and it may be rigorous, but it makes no sense to say it is true in an absolute sense. We can know that Popper's Realism is not endorsed by metaphysics but is shown to be logically indefensible, but this does not prove it is false.
These issues can mostly be bypassed if we say that truth is knowledge and that we cannot know truth unless we know it is truth. Thus for the mystical epistemologist truth is what we know is true. This approach considerably simplifies the issues.
The arguments of Kant and Popper work well for scientific and metaphysical theories. These are never true or false. But their arguments say nothing about whether a person can know metaphysical truths. It is simply taken for granted that they cannot.
The physicist Paul Davies has a more informed view. From his analysis he
concludes that Popper and Kant et al are correct as to the limits of theoretical knowledge and its inability to ever become truth or certain knowledge, so takes the view that if there is a way to know metaphysical truths it could only be by way of the methods described in mysticism. Their arguments do not apply to direct unmediated knowledge of Reality so although he is sceptical he does not have an argument to defeat mysticism's knowledge claims. This makes his Mind of God an unusual and brave book.
Was Kant mistaken? His insight has been widely acclaimed by professional philosophers for two centuries…no refutation of his work has been widely accepted as cogent.
I'm not sure what insight you mean. Is it that we cannot know the 'thing-in-itself'? This is said in Upanishads. The knower cannot know the knower but can only be it. Who is there to know the knower or understand the understander?
Kant does not prove we can know nothing of the thing-in-itself (even if he proved there is one) so his insight is a conjecture.
Was Popper mistaken? If so, why? If not, is it more productive when engaging in philosophical inquiry to publicly examine the justifications for our beliefs than to insist or to imply that they’re indisputable?
Beliefs as we usually define them are always disputable. They are never truth or knowledge. For truth and knowledge one has to go beyond beliefs. Popper is correct, it seems to me, in most of what he says, but he has a limited view of knowledge that means his arguments only work for physical and metaphysical theories. He says nothing about whether knowledge of truth is possible thus leaves us free, as Davies notes, to speculate that we can know truths and know we know them.
Russell's tradition is muddled on this issue, as we see from his confession that he is unable acquire any knowledge from metaphysics and has no idea how human beings know things. This problem is shared by his tradition and it therefore has a confused approach to knowledge and truth.
What are the reasons for applying fallibilism? What are the reasons to believe that certainty is a better approach?
If fallibilsm states that certainty is impossible then it is a speculative theory with no proof or evidence to support it and much counter-evidence. If it is the view that we should never be dogmatic about our speculations then it is just good philosophcial practice.
The search for certainty requires that we acknowledge the fallibility of our speculative theories and seek more certain knowledge, so the two approaches are complementary to some extent. In this sense fallibilism is just Cartesian doubt. But why would anyone do philosophy unless they think they can reach certainty?
If it is possible for human being to know metaphysical truths in the way mysticism teaches then the arguments of Popper and Kant would still be sound in respect of our speculative theories. That is, they are sound if we believe knowledge of truth is impossible. They are not global arguments though, since they do not rule out the possibility of such knowledge.
For this reason I see Davies as having a better grasp of the issues than either of them. He knows that he doesn't know whether such knowledge is possible.
Truth is knowledge. As far as we'll ever know there is no such thing as truth in the absence of knowledge. A truth is what we know to be true. 'Justified true belief' does not qualify, and coherence or correspondence are just useful conditions for building theories.
Meanwhile, it is perfectly possible to know relative truths. For instance, we can know that a speculative theory is never a known truth.
This is off-the-cuff so is not well organised but perhaps it outlines some of the issues that arise for the question.