Periodicity refinements for Romik's sequence $d(n)$ modulo powers of primes $p\equiv1\pmod4$

Let $p$ be a prime with $p\equiv1\pmod4$, let $e\ge1$, and let $(d(n))_{n\ge1}$ be the sequence defined in the paper. The fourth conjecture. (1) The sequence $d(n)$ modulo $p^e$ is eventually periodic, with a not necessarily minimal period length

$$\frac18p^{e-1}(p-1)^2.$$

(2) There exists a constant $C_{p,e}$ such that

$$d\left(n+\frac{p^{e-1}(p-1)}4\right)\equiv C_{p,e}d(n)\pmod{p^e}$$

for all $n\ge1$, and

$$C_{p,e}^{(p-1)/2}\equiv1\pmod{p^e}.$$

The conjecture refines the paper's periodicity theorem. The authors note that computer data suggest pure periodicity, but deliberately conjecture only eventual periodicity; item (2) strengthens and generalises an earlier conjecture attributed to Wakhare.