Subpolynomial growth of the last near-kth-power index

Prove that for every integer k > 1 and every real ε > 0, the quantity M_k(d) = max{n : A_k(n) ≤ d} satisfies M_k(d) = o(d^ε) as d tends to infinity, together with the asserted identity M_k(0) = 1.

Background

The function M_k(d) records the largest index n for which p(n) lies within distance d of a kth power. Conjecture 1 would imply that this maximum is finite for each fixed d.

Conjecture 2 refines the finiteness claim by predicting extremely slow growth in d: M_k(d) should be bounded by every positive power of d. Its first part is equivalent to Sun’s conjecture at d = 0.