Integrality and irreducibility conjecture for level-$N$ class polynomials

Let $D$ and $N$ be as in the source, let $m\in\mathcal{M}_{D,N}$, and let $j_N^+$ be a Hauptmodul for $\Gamma_0^+(N)$. Define

$$H_{D,N,m}(X)=\prod_{Q\in\mathcal{Q}_{D,N,m}^0/\Gamma_0^+(N)}\left(X-j_N^+(\tau_Q)\right).$$

Class-polynomial conjecture. The polynomial $H_{D,N,m}(X)$ belongs to $\mathbb{Z}[X]$ and is irreducible. This is proposed as an analogue of the preceding classical class-polynomial theorem; the source gives no resolution of the assertion.