Supercongruence conjecture for hypercubic, hyper-octahedral and simplicial families

Let $F_N(t)=\sum_{n=0}^{N-1}f_nt^n$ be the truncation of $F(t)$ at $t^N$, and let $t\mapsto t^\sigma$ be the excellent Frobenius lift. The congruence

$$\frac{F(t)}{F(t^\sigma)}\equiv \frac{F_{mp^s}(t)}{F_{mp^{s-1}}(t^\sigma)}\mod{p^s}$$

holds for every $m,s\geq 1$. Supercongruence conjecture. The same congruence holds modulo $p^{2s}$ for $m=2$ in the hypercubic and hyper-octahedral families, and for $m=n+1$ in the simplicial families. Numerical experiments support this strengthening, but it has not been proved for any example of $g$ in the source.