What do you mean by "when the truth-value of the conjecture is unknown"? That neither C nor ¬C is in your bound agent's store of known theorems?

I meant that the agent is in some state of uncertainty. I'm trying to contrast the case where we are more certain of either C or ¬C with that where we have a significant degree of uncertainty.

If I know C is false, the purported proofs are trivially independent because they're fully determined incorrect?

Yeah, this is just the trivial case.

Why is S1 independent of S2 given C is true?

I was talking about the simple case where there are no other causal links between the two proofs, like common lemmas or empirical observations. Those do change the causal structure by adding extra nodes and arrows, but I was making the simplifying assumption that we don't have those things.