Infinitely many distinct positive-half-plane roots

Establish whether the D’Arcais polynomials \(R_n^\sigma(X)\) possess infinitely many pairwise distinct roots in the open positive half-plane.

Background

The paper proves that a positive proportion of the reduced D’Arcais polynomials $R_n^\sigma(X)=P_n^\sigma(X)/X$ are not Hurwitz, meaning that each such polynomial has at least one root in the open right half-plane. However, the authors do not establish whether these roots are infinitely numerous and pairwise distinct across the family. The problem asks for a stronger global description of the positive-half-plane roots.