Heim–Neuhauser's log-concavity challenge for D'Arcais polynomials

Let $P_n^\sigma(X)=\sum_{k=0}^n p_n^\sigma(k)X^k$ be the D'Arcais polynomial defined by

$$\prod_{m=1}^\infty(1-q^m)^{-X}=\sum_{n=0}^\infty P_n^\sigma(X)q^n,$$

and call a sequence $(a_k)_{k\in\mathbb{N}}$ log-concave at $k\geq 1$ when $a_k^2\geq a_{k-1}a_{k+1}$. Heim–Neuhauser's challenge. For every $n\in\mathbb{N}$, the sequence $(p_n^\sigma(k))_{k\in\mathbb{N}}$ is log-concave at each $k\geq 1$. This is a proposed coefficient-growth property for the D'Arcais polynomials; the source supplies no resolution status.