Heim–Neuhauser conjecture for D’Arcais polynomials

Let $P_0(x)=1$ and, for $n\geq 1$, define the D’Arcais polynomials by

$$P_n(x):=\frac{x}{n}\sum_{j=1}^n\sigma(j)P_{n-j}(x),\qquad \sigma(j):=\sum_{d\mid j}d.$$

For integers $a>b\geq 0$, define

$$\Delta_{a,b}(x):=P_{a-1}(x)P_{b+1}(x)-P_a(x)P_b(x).$$

Heim–Neuhauser conjecture. For all $x\geq 2$,

$$\Delta_{a,b}(x)\geq 0,$$

except for $b=0$ and $(a,b)=(6,4)$. The inequality remains true for $x\geq 3$ when $b=0$, and for $x\geq x_{6,4}$ when $(a,b)=(6,4)$, where $x_{a,b}$ is the largest real root of $\Delta_{a,b}(x)$. This conjecture extends the colored-partition inequality to D’Arcais polynomials; the source does not provide a resolution status.