A proof of the conjecture would essentially just move the situation from "we think there could be counterexamples, but so far we haven't found any" to "there really are no counterexamples anywhere, you can stop looking". The interesting thing here would be the explanation why there can be no counterexamples, which is exactly the thing that the proof wouldn't give you.

On the other hand, a counterproof would either directly present the community with a counterexample - and maybe reveal some interesting class of objects that was overlooked so far - or at least establish that there have to be counterexamples somewhere, which would probably give more motivation to efforts of finding one.

(Generally speaking here. I'm not a mathematician and I can't really say anything about the hypotheses in question)