3 problems

Partition-theoretic stabilization conjecture. For each non-negative integer $d$, there is a positive integer $N_d$ such that for $k\geq N_d$,

Refinement of the perfect-power repulsion conjecture. If $k>1$, then $M_k(0)=1$, and for every $\varepsilon>0$,

Perfect-power repulsion conjecture. If $k>1$ and $d\geq 0$, then there are at most finitely many $n$ for which