Motzkin numbers nonvanishing modulo the squares of 31, 37, and 61

Let $M(n)$ denote the $n$th Motzkin number, and let $p$ be one of the primes in the set

$$\{31,37,61\}.$$

Nonvanishing conjecture. For all $n\geq 0$,

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

The conjecture is suggested by experimental evidence, but the automata for these moduli had not been computed in the source, so its status remains open.