I realize that I am not sure how you use the word “conjecture”. Here are two sample readings of this word in your “Hod Conjecture”. (I realize this does not exhaust the possibilities.)
A) It is provable in ZFC that if $kappa$ is a huge cardinal, then the HOD conjecture holds in .
B) It is simply true that the HOD conjecture holds. No implication concerning provability is intended.
(I realize position B is incomprehensible (and/or absurd) to the Friedman brothers.) My position that CH is false (and that is weakly inaccessible) is much like this suggested alternative B.