Higher-level supercongruence for truncated hypergeometric series

Fix an integer $d\ge2$, let $p$ be an odd prime, and define

$$F_{p^s}(z)=\sum_{n=0}^{p^s-1}\left({(1/2)_n\over n!}\right)^d z^n$$

and

$$f_p(z)=\lim_{s\to\infty}{F_{p^s}(z)\over F_{p^{s-1}}(z)}.$$

Let $\epsilon=\pm1$ and suppose that $F_p(\epsilon)$ is a $p$-adic unit. Higher-level supercongruence. For every integer $s\ge1$, one should have

$$F_{p^s}(\epsilon)\equiv f_p(\epsilon)F_{p^{s-1}}(\epsilon)\pmod{p^{2s}}.$$

This is proposed as a generalization of the paper's theorem, which gives the corresponding first-level modulus-$p^2$ congruence. The assertion is supported by numerical experiments and remains conjectural.