D'Arcais polynomial positivity conjecture

Let $P_n^{\sigma}(x)$ denote the $n$th D'Arcais polynomial associated with the divisor-sum function $\sigma$, and define

$$\Delta_n^{\sigma}(x):=(P_n^{\sigma}(x))^{n+1}-(P_{n+1}^{\sigma}(x))^n.$$

D'Arcais positivity conjecture. For every integer $n\geq 6$ and every real $x\geq 1$,

$$(P_n^{\sigma}(x))^{n+1}-(P_{n+1}^{\sigma}(x))^n>0.$$

At $x=1$, this specializes to Sun's conjecture for partition numbers. The supplied material does not state whether this broader polynomial claim has been resolved.