Beukers' 5-adic valuation conjecture for Apéry numbers

For all $n\in\mathbb{N}$, define

$$n=\sum_{k=0}^N n_k5^k,\qquad n_k\in\{0,\dots,4\}.$$

Let $\alpha$ be the number of indices $k\in\{0,\dots,N\}$ such that $n_k\in\{1,3\}$. Here

$$A_1(n):=\sum_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$

Beukers' conjecture. The integer $5^\alpha$ divides $A_1(n)$.

This is one of Beukers' conjectures on the $p$-adic valuations of Apéry numbers, refining the divisibility information supplied by the 5-Lucas property. The paper states that these conjectures are proved there.