The minimal-polynomial degree conjecture for recursive sequences modulo prime powers

Let $p$ be prime and let $p^\gamma$, with $\gamma\ge1$, be a modulus. A minimal polynomial is a polynomial of least degree giving the relevant recursive-sequence relation modulo $p^\gamma$. Minimal-polynomial degree conjecture. The degree of a minimal polynomial for the modulus $p^\gamma$ is the least $d$ such that

$$v_p(d!)\ge\gamma.$$

This conjecture is proposed from the minimal-polynomial calculations for the moduli $p,p^2,\ldots,p^{p+1}$. The source gives no resolution or further evidence for its status.