My belief is that epistemic closure is false. My argument relies on mathematics. In mathematics, one may know the axioms of number theory or set theory as true, but one probably doesn't know all the theorems of number theory or set theory. Is this a good argument?
It has an essential truth but lacks subtlety.
Step down the level of complexity to one axiom -- succession. Do you believe the natural numbers exist? They are the 'logical closure' of counting by applying the axiom of succession. Obviously, we will never reach certain numbers, and to that degree, those numbers have never been real. But in a more abstract way, every possible number is just as real as 'two'.
The neo-Intuitionist approach here is to weaken our reliance upon negation in a way that splits the meaning of 'exist'.
One can admit that those things are potential steps in some abstract process and that any decision one makes about that process and the system that supports it as a whole must not contradict the idea of any one of them occurring. This means they "do not not-exist": one must not risk contradicting the assumption of their existence, or we will abandon our intuitive traction on the remaining reality.
But at the same time, one should deny that they really can be encountered in reality. So they "do not exist".
Since Intuitionism denies meaning to the law of the excluded middle in infinite domains, things are free to not exist, yet not not-exist.
The whole of mathematics then falls into the realm of things that do not not-exist. We cannot contradict any potentially valid deduction and get away with it. But those theorems that will never be proven (of which there are obviously many) do not exist.