Positivity conjecture for subgroup-counting D'Arcais differences

Let $g_{\ell}(n)$ be the number of subgroups of $\mathbb{Z}^{\ell}$ of index $n$, and let $\Delta_n^{g_{\ell}}(x)$ denote the associated D'Arcais difference. Set

$$a_3=8,\qquad a_4=8.$$

Subgroup-counting positivity conjecture. If $\ell=3$ or $\ell=4$, then for every $n\geq a_{\ell}$ and every real $x\geq 1$,

$$\Delta_n^{g_{\ell}}(x)>0.$$

The claim is supported in the supplied material by computations for $8\leq n\leq25$ and plots; no proof or disproof is given.