Apéry-like coefficient formulas for squares of level-5 and level-7 supersingular polynomials

For a prime $p$, let $ss^{(5*)}_{p}(X)$ and $ss^{(7*)}_{p}(X)$ be the level-5 and level-7 supersingular polynomials, and let $u_{5}^{*}(n)$ and $u_{7}^{*}(n)$ be the Apéry-like sequences defined in the source. Let $m_{5}$ and $\mu_{5}$, and $m_{7}$ and $\mu_{7}$, be the quantities defined in the preceding level-5 and level-7 formulas.

Apéry-like supersingular-polynomial conjecture. For every prime $p\geq7$,

$$ss^{(5*)}_{p}(X)^{2}=(X^{2}-44X-16)^{\mu_{5}}\sum_{n=0}^{2(m_{5}+\mu_{5})}u_{5}^{*}(n)X^{2(m_{5}+\mu_{5})-n}\pmod p.$$

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

$$ss^{(7*)}_{p}(X)^{2}=(X+1)^{\mu_{7}}(X-27)^{\mu_{7}}\sum_{n=0}^{2(m_{7}+\mu_{7})}u_{7}^{*}(n)X^{2(m_{7}+\mu_{7})-n}\pmod p.$$

These conjectural identities connect supersingular polynomials with Apéry-like sequences arising from Heun representations of Eisenstein series. The source gives no resolution evidence.