Heim–Neuhauser's Hurwitz conjecture for D'Arcais polynomials

For $n\geq 1$, let $P_n^{\sigma}(x)$ be the D'Arcais polynomial defined by

$$\prod_{m=1}^{\infty}(1-q^m)^{-x}=\sum_{n=0}^{\infty}P_n^{\sigma}(x)q^n.$$

A polynomial is Hurwitz if the real parts of all its zeros are negative. Heim–Neuhauser's conjecture. The polynomial $P_n^{\sigma}(x)/x$ is Hurwitz, and all its zeros are simple. The conjecture concerns the location and multiplicity of the zeros of D'Arcais polynomials; the supplied source gives no evidence that it has been resolved.