Relative-primality conjecture for the $D_m$ auxiliary polynomials

Let $h_{p,m}(t)$ and $\tilde{h}_{p,m}(t)$ be the polynomials defined by the factorizations of the residue polynomials $H_{p,m}(t)$ and $\tilde{H}_{p,m}(t)$ for the $D_m$ orbifold vertex algebra. Relative-primality conjecture. The polynomials $h_{p,m}$ and $\tilde{h}_{p,m}$ are relatively prime. This condition is used to obtain polynomial relations in Zhu's algebra and hence bounds on its dimension; the source gives no resolution beyond computations for small values of $(m,p)$.