Class-number formula for the number of linear factors at general levels

Let $p\geq5$ be prime, let $N\in\mathfrak{S}\setminus\{2\}$, and assume $N\neq p$. Let $L^{(N*)}(p)$ be the number of linear factors of $ss_{p}^{(N*)}(X)$, $L(p)$ the level-1 quantity, and $h(\sqrt{-Np})$ the class number appearing in the source. The Kronecker symbol $\left(\frac{\cdot}{Np}\right)$ is the product of the corresponding Legendre symbols.

General-level linear-factor formula conjecture.

$$L^{(N*)}(p)=\frac12\left(1+\left(\frac{-p}{N}\right)\right)L(p)+\frac18\left\{2+\left(1-\left(\frac{-1}{Np}\right)\right)\left(2+\left(\frac{-2}{Np}\right)\right)\right\}h(\sqrt{-Np}).$$

This conjecturally expresses the number of linear factors through the level-1 count and an imaginary quadratic class number. The source gives no resolution evidence.