Strengthening the Catalan–Larcombe–French congruence modulo p^3

Establish the conjectured strengthening modulo p^3 of the congruence relating the sum of squared Catalan–Larcombe–French numbers to the fourth-power central-binomial-coefficient sum: for every prime p>3, determine whether the congruence \(\sum_{k=0}^{p-1}\mathcal{P}_k^2/128^k\equiv(-1/p)\sum_{k=0}^{p-1}\binom{2k}{k}^4/256^k\pmod{p^3}\) holds.

Background

The paper proves a congruence modulo $p^2$ connecting the Catalan–Larcombe–French numbers $\mathcal{P}_k$ with the fourth powers of central binomial coefficients. The authors then report numerical evidence that the same congruence may hold one power of p more strongly, modulo $p^3$.

The unresolved issue is therefore whether the observed numerical strengthening is valid for every prime $p>3$, rather than merely for the cases tested computationally.