Quattrini's Fourier coefficient formula for Cohen–Eisenstein series

Let $B$ be a definite quaternion algebra ramified at exactly one finite prime $p$, and let $N=pM$ with $p mid M$ be square-free. Write

$$\mathcal{H}=\sum_{i=1}^{n}\frac{1}{w_i}g_i=\sum_{i=1}^{n}\frac{1}{2w_i}+\sum_{D>0}\mathcal{H}(D)q^D.$$

For $D\in\mathbb{N}$, let $-D$ be a fundamental discriminant, and define $h(-D)$ as the cardinality of the ideal class group of the ring of integers of $\mathbb{Q}(\sqrt{-D})$, $u(-D)$ by letting $2u(-D)$ be the cardinality of its unit group, and let $s(D)$ denote the number of prime divisors of $M$ at which the relevant quadratic character takes the value $-1$. Quattrini's Fourier coefficient conjecture. If

$$\left(\frac{-D}{p}\right)\neq 1$$

and

$$\left(\frac{-D}{q}\right)\neq -1$$

for every prime $q\mid M$, then

$$\mathcal{H}(D)=\frac{2^{\omega(N)-1-s(D)}h(-D)}{u(-D)}.$$

This conjecture predicts an explicit class-number formula for the Fourier coefficients of the Cohen–Eisenstein series attached to a definite quaternion algebra. It was proposed from observed congruences among weight-two modular forms and Brandt-matrix eigenvectors; the supplied source gives no resolution evidence, so its status is left open.