Yes, I think the only way to determine why the statement is true is by using other statements, even if in different form if talking about the nature of these statements (I can explain visually a statement that comes from discourse) but I think we have no escape because these statements we're trying to prove exists in a context that generated them by other statements. Telling why - so proving - is acting to unveil the background statements that made the statement that had to be proven emerge.
So I think that objective reality does not comprehend proofs because this is a concept that lies in our subjective perspective that is proper among human beings.
You were asking:
Does this mean that objective truth/reality can not be justifiably
said to exist?
I found it a pretty interesting question even because it takes into account faith.
Well, objective reality exists but it does not comprehend truth or lack of it. Objective reality have to exist and work because we are its products, but "objective truth" seems to me like a contradictory concept because we have not that control over the whole we're part of.
Thanks you for this question, it allowed new questions to rise on me.