Positivity conjecture for the plane-partition D'Arcais differences

Let $\sigma_2(n):=\sum_{d\mid n}d^2$, and let $\Delta_n^{\sigma_2}(x)$ denote the difference associated with the D'Arcais polynomials for $g(n)=\sigma_2(n)$. Plane-partition positivity conjecture. For every $n\in\mathbb{N}$,

$$\Delta_n^{\sigma_2}(x)>0\quad\text{for }x\geq g(2)=5,$$

and, moreover, for every $n\geq 6$,

$$\Delta_n^{\sigma_2}(x)>0\quad\text{for }x\geq 1.$$

The claim is motivated by the computed values and plots in the paper; the supplied material gives no resolution, so its status remains open.