Abdesselam's log-concavity conjecture for subgroup-counting polynomials

Let $g_\ell(n)$ denote the number of subgroups of index $n$ in $\mathbb{Z}^\ell$, and write

$$P_n^{g_\ell}(X)=\sum_{k=0}^n p_n^{g_\ell}(k)X^k.$$

Call a sequence $(a_k)_{k\in\mathbb{N}}$ log-concave at $k\geq 1$ when $a_k^2\geq a_{k-1}a_{k+1}$. Abdesselam's conjecture. For every $\ell,n\in\mathbb{N}$, the sequence $(p_n^{g_\ell}(k))_{k\in\mathbb{N}}$ is log-concave at each $k\geq 1$. This generalizes the Heim–Neuhauser challenge, since $g_2=\sigma$; the source supplies no resolution status.