Congruence behavior of the matrix $R(n,k)$

Let $R(n,k)$ be the matrix introduced in the paper, and let $R^{-1}(n,k)$ denote its inverse matrix. The fifth conjecture. (1) For fixed $k$ and any prime power $p^e$ with $p\equiv1\pmod4$, the sequence $(R(2n+k,k))_{n\ge0}$ is purely periodic modulo $p^e$.

(2) For fixed $k$ and any prime power $p^e$ with $p\equiv3\pmod4$, the sequence $(R(2n+k,k))_{n\ge0}$ is eventually zero modulo $p^e$.

(3) For fixed $a$ and any power $2^e$, the sequence $(R^{-1}(2n+k,k))_{k\ge0}$ is periodic modulo $2^e$, with period length $2^{e-3}$.

These are conjectural congruence properties of the matrix; item (3) is described as a strengthening of a theorem specialised to the sequence $y(k)=u(k)$.