Apéry-like coefficient formula for class 3C supersingular polynomials

Let $p\geq5$ be prime, let $ss_{p}^{(3C)}(X)$ be the class-3C supersingular polynomial, and let $\varepsilon$ be the parameter used in its degree formula.

Class 3C Apéry-like conjecture.

$$ss^{(3C)}_{p}(X)^{2}=\{(X-12)(X^{2}+12X+144)\}^{\varepsilon}$$ $$\times\sum_{n=0}^{(p-1)/2}\binom{2n}{n}\left\{\sum_{k=0}^{[n/3]}(-3)^{n-3k}\binom{2k}{k}\binom{3k}{k}\binom{n}{3k}\right\}X^{(p-1)/2-n}\pmod p.$$

This conjecturally relates the square of the class-3C supersingular polynomial to an Apéry-like sequence. The source gives no resolution evidence.