Askey–Gasper rational-function coefficient supercongruence

Let

$$F(x_1,x_2,x_3)=\frac{1}{1-(x_1+x_2+x_3)+4x_1x_2x_3}$$

and write its Taylor expansion as

$$F(\boldsymbol{x})=\sum_{\boldsymbol{n}\in\mathbb{Z}_{\geqslant 0}^3}G(\boldsymbol{n})\boldsymbol{x}^{\boldsymbol{n}}.$$

This is the Askey–Gasper rational function. Askey–Gasper coefficient supercongruence. For every prime $p\geqslant 5$ and integer $r\geqslant 1$,

$$G(p^r\boldsymbol{n})\equiv G(p^{r-1}\boldsymbol{n})\pmod{p^{3r}}.$$

The conjecture is motivated by numerical evidence for a rational function on the boundary of positivity and extends the supercongruence phenomenon from diagonal Apéry-like sequences to all coefficients.