The higher Almkvist-Zudilin supercongruence

Let $\mathbb{N}^+$ denote the positive integers, and for integers $i\geq 0$ and $n\geq 1$ define

$$a_i(n)=\sum_{k=0}^{\lfloor(n-i)/3\rfloor}(-1)^{n-k}\binom{3k+i}{k}\binom{2k+i}{k}\binom{n}{3k+i}\binom{n+k}{k}3^{n-3k-i}.$$

Let $p$ be a prime satisfying $p>2i$. The higher Almkvist-Zudilin supercongruence. For $n,i\in\mathbb{N}^+$,

$$a_i(pn)\equiv_{p^3}(-1)^{i-1}\frac{a_1(pn)}{i^2\binom{2i-1}{i-1}}\equiv_{p^3}\frac{(-1)^{i-1}p^2\binom{n+2}{2}a_1(n)}{i^2\binom{2i-1}{i-1}}.$$

This extends the paper's congruence program from $a_0(n)$ to the sequences $a_i(n)$ with positive index. The supplied context gives proof outlines only for the case $i=1$ and says that the remaining cases are left to the reader, so the general assertion remains open in the source.