Resultant multiplicity conjecture for supersingular polynomials at levels 2, 3, 5 and 7

Let $p\geq5$ be prime with $p\neq N$. For each relevant level $N$, let $R_{N}(X,Y)$ be the algebraic relation used to define the modular parameter, and set

$$V_{p}^{(N*)}(Y)=\operatorname{Resultant}_{X}[ss_{p}(X),R_{N}(X,Y)].$$

Let $\delta$, $\varepsilon$, $\nu$, and $\mu_{5}$, $\mu_{7}$ have the meanings used in the source.

Resultant multiplicity conjecture. The following congruences hold modulo $p$:

$$Y^{\varepsilon}(Y-256)^{\nu}V_{p}^{(2*)}(Y)=(Y+144)^{2\delta}(Y-648)^{\varepsilon}ss_{p}^{(2*)}(Y)^{2},$$ $$Y^{\delta}(Y-108)^{\delta}V_{p}^{(3*)}(Y)=(Y+192)^{2\delta}(Y^{2}-576Y-1728)^{\varepsilon}ss_{p}^{(3*)}(Y)^{2},$$ $$(Y^{2}-44Y-16)^{\mu_{5}}V_{p}^{(5*)}(Y)=(Y^{2}+216Y+144)^{2\delta}(Y^{2}-540Y-6480)^{\varepsilon}ss_{p}^{(5*)}(Y)^{2},$$ $$(Y+1)^{\mu_{7}}(Y-27)^{\mu_{7}}V_{p}^{(7*)}(Y)=(Y^{2}+224Y+448)^{2\delta}(Y^{4}-528Y^{3}-9024Y^{2}-5120Y-1728)^{\varepsilon}ss_{p}^{(7*)}(Y)^{2}.$$

The conjecture predicts that the resultant has supersingular roots with multiplicity essentially two, apart from explicitly identified exceptional factors. The source gives no resolution evidence.