Rowland–Yassawi's conjecture on Motzkin numbers modulo prime squares

Let $M(n)$ denote the $n$th Motzkin number. The question concerns primes $p$ for which $p^2$ never divides any Motzkin number.

Rowland–Yassawi's conjecture. For each $p\in\{31,37,61\}$ and every $n\in\mathbb{Z}_{\geq 0}$,

$$M(n)\mathrel{\not\equiv}0\pmod{p^2}.$$

This is part of the open question of whether infinitely many primes have this property. The primes $5$ and $13$ are already known to have it, while the assertion for $31$, $37$, and $61$ remains open in the supplied source.