Heun polynomial representations for supersingular polynomials at levels 5 and 7

Let $p$ be prime. For $N=5$, define $m_{5}$, $\mu_{5}$, $\phi$, and $Hl_{5}(X)$ as in the displayed formulas below; for $N=7$, define $m_{7}$, $\mu_{7}$, and $Hl_{7}(X)$ similarly.

$$m_{5}=\frac{p-1}{4}+\frac14\left(1-\left(\frac{-1}{p}\right)\right)-\mu_{5},\quad \mu_{5}=\frac12\left(1-\left(\frac{-5}{p}\right)\right),\quad \phi=\frac{1+\sqrt5}{2},$$ $$Hl_{5}(X)=Hl\left(-\phi^{10},-\frac{(22\mu_{5}+3)\phi^{5}}4;-m_{5},\mu_{5}+\frac12+\frac14\left(\frac{-1}{p}\right),1,\mu_{5}+\frac12;\frac{4\phi^{5}}X\right),$$

and define $m_{7}$, $\mu_{7}$, and $Hl_{7}(X)$ by the corresponding formulas in the claim.

Heun polynomial representation conjecture. For every prime $p\geq7$,

$$ss_{p}^{(5*)}(X)=X^{m_{5}}(X^{2}-44X-16)^{\mu_{5}}Hl_{5}(X)\pmod p.$$

For $p=5$ and $p\geq11$,

$$ss_{p}^{(7*)}(X)=X^{m_{7}}(X+1)^{\mu_{7}}(X-27)^{\mu_{7}}Hl_{7}(X)\pmod p.$$

These formulas conjecturally extend the hypergeometric descriptions of supersingular polynomials to levels 5 and 7, where Heun polynomials replace hypergeometric polynomials. The source supplies no resolution evidence.