“p is true/fals regardless of any conscious experience, which means p always has a truth-value.”
Can't you respond to that in the following way?
Sure, a statement is indeed determinately true regardless of whether we know it to be true or how we know it's true... So what? That determinate truth-value doesn't have any epistemic or even metaphysical point.
What I mean by that is if we can't establish a procedure for determining its truth, then what purpose does the locution “p is determinately true or false” serve?
That realist's truth is “a wheel in the mechanism that doesn't have a function”, as Wittgenstein put it (though about something else).
I think it was Bertrand Russell (or Michael Dummett) who made this statement:
“There's a flying teapot floating around a distant star X in a distant galaxy Y.”
Now there's no way of determining the truth or falsehood of this statement. (It didn't help matters by saying it's a teapot.) Still, the statement, “There is a teapot floating around star X in galaxy Z”, is either true or false – determinately.
p is determinately true or false either way. However, what matters is how we determine its truth. What doesn't seem to matter is that it's determinately true or false regardless; especially since we don't know its truth-vale and, therefore - from the realist's perspective - that may be the end of the story.
The anti-realist can say that a proposition is “truth-apt” now; though it can have its precise truth-value determined in the future. However, the realist will still say that it's both truth-apt now; as well as true now. Later, the mathematician or philosopher may come to determine its truth (or falsity); though it's still true (or false) now.
The case of Fermat's Last Theorem, for example, adds more difficulties to this. Though the point of this example is – is it? - that the proof brought/brings the theorem's truth into existence (as it were). This is what the realist disputes. (Does he?) The intuitionists (or constructivists) would have said that Fermat's theorem had no truth until it was/is proved. Thus truth = proof.
The problem here is that p may be determinately true or false when it comes to the case of the flying teapot; though not determinately true or false when it comes to an unproven (or any) mathematical statement or theorem.
I've never really seen the anti-realist rejection of bivalence as, well, a genuine rejection. What I mean by that is that to argue that there's a third truth-value (which is indeterminate) isn't really a rejection of a proposition's being determinately true or false. Thus even if the truth value of p is indeterminate, that simply tells us about our epistemic situation. It does have an indeterminate truth-value for us. However, it's still determinately true or false (though for whom - for no one?).