Degree and linear-factor conjecture for class 3C supersingular polynomials

Let $p\geq5$ be prime. Let $ss_{p}^{(3C)}(X)$ be the supersingular polynomial associated with the group $\Gamma_{0}(3|3)$, and let $L^{(3C)}(p)$ be its number of linear factors. Let $L(p)$ denote the number of linear factors of the level-1 supersingular polynomial.

Class 3C supersingular-polynomial conjecture.

$$\deg ss_{p}^{(3C)}(X)=\frac{p-1}{4}+\frac32\varepsilon,\qquad L^{(3C)}(p)=\left(2+\left(\frac{-3}{p}\right)\right)L(p).$$

This conjecture predicts both the degree and the number of linear factors for the class-3C supersingular polynomial. The source gives no resolution evidence, and the supplied context does not define $\varepsilon$.