The initial log-concavity thresholds for k-th power partition functions

For an integer $k\geq 2$, let $p^k(n)$ denote the number of partitions of $n$ into perfect $k$-th powers. Define $N_k$ to be the smallest index such that the sequence $\{p^k(n)\}$ is log-concave for all $n>N_k$.

Initial log-concavity threshold conjecture. For $k\leq6$, the smallest such values are

$$\begin{array}{c|ccccc} p^k(n)&p^2(n)&p^3(n)&p^4(n)&p^5(n)&p^6(n)\\ \hline N_k&1041&15655&637854&2507860&35577568 \end{array}$$

These values record the first index after which log-concavity is observed for the square through sixth-power partition functions. The source gives no proof or resolution of this finite-data assertion.