Level-54 attractor-point supercongruence conjecture

For every prime $p\ne3$, consider the truncated hypergeometric sum

$$\sum_{k=0}^{p-1}\binom{-1/3}{k}^2\binom{-2/3}{k}^2(-1/8)^k.$$

Let $a_p$ be the coefficient of $q^p$ in the weight $4$ cusp form of level $54$ beginning $q+2q^2+4q^4+3q^5+\cdots$. Level-54 attractor-point supercongruence conjecture. For every prime $p\ne3$,

$$\sum_{k=0}^{p-1}\binom{-1/3}{k}^2\binom{-2/3}{k}^2(-1/8)^k\equiv a_p\pmod{p^3}.$$

This conjecture predicts a modular-form congruence at the attractor point $t_0=-1/8$ for the indicated hypergeometric function. The source reports it as experimental and gives no resolution.