Beukers' valuation conjecture for the second Apéry sequence

Let $p$ be a prime number satisfying

$$p\equiv 3\pmod 4.$$

For $n\in\mathbb{N}$, write

$$n=\sum_{k=0}^N n_kp^k,\qquad n_k\in\{0,\dots,p-1\}.$$

Let $\alpha$ be the number of indices $k\in\{0,\dots,N\}$ such that $n_k=\frac{p-1}{2}$. Define

$$A_2(n):=\sum_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}.$$

Beukers' conjecture. The integer $p^\alpha$ divides $A_2(n)$.

This conjecture strengthens the known congruence $A_2((p-1)/2)\equiv0\pmod p$ for primes $p\equiv3\pmod4$, using the $p$-Lucas property of the second Apéry sequence. The paper states that Beukers' conjectures on Apéry numbers are proved there.