In order for someone to make a conjecture that A does B, and for someone else to point out that A actually doesn't do B, both parties need to agree on the meanings of A and B, as well as what it means for A to do B, usually by appealing to C, an independent common ground. You climb down as many levels of abstractions as needed until you reach C, because otherwise you're just talking past each other.
Even in a coherentist system, the network of existing beliefs forms the common ground against which questionable beliefs are tested, and some beliefs are held firmer than others. Those beliefs are treated as already justified for the purpose of the current investigation. Few people who talk about consciousness would ever consider denying that they are conscious.
> In order for someone to make a conjecture that A does B, and for someone else to point out that A actually doesn't do B, both parties need to agree on the meanings of A and B, as well as what it means for A to do B…
You try to establish common ground, of course, but there’s no process you can follow that guarantees that you’re not being misunderstood or justifies your belief that you’re not being misunderstood.
However, I believe there is nothing axiomatic about the existence of consciousness: not only is consciousness not “one localized thing” but “a collection of delocalized features”, but there is nothing trivial about it, and I doubt very much that everyone in the room would agree on a single definition of it without getting into semantic quabbles.
Axioms tend to be much simpler.
If, on the other hand, the pursuit of knowledge consists of something like using creativity and reason to solve problems by making new conjectures and criticizing them, then no infinite regress or un-criticizable is required. According to ideas like this, the goal is to solve problems rather than to “justify my beliefs” or “increase my confidence” or “guarantee that I’m not incorrect.”