Modulo-$p^3$ conjecture for products of truncated hypergeometric series

Let $d$ satisfy the hypotheses in Theorem $\mathrm{main}$, put ${\bf b}=(\frac12,\frac{1}{d},\frac{d-1}{d})$, and let $\widetilde{E}_d((1-\sqrt{1-\lambda})/2)$ be the associated elliptic curve. Let $[G]_{p-1}(\lambda)$ denote truncation at degree $p-1$, and let $u_p$ be the unit root of geometric Frobenius at $p$ acting on the first cohomology of this elliptic curve. The constants are $k_2=k_6=-1$, $k_3=-3$, and $k_4=-2$.

Product supercongruence conjecture. For primes $p\geq 5$,

$$[F_{0,{\bf b}}\cdot F_{2\alpha_N,{\bf b}}]_{p-1}(\lambda)\equiv \begin{cases} 0 & \text{if }\widetilde{E}_d\left(\frac{1-\sqrt{1-\lambda}}{2}\right)\text{ is supersingular at }p,\\ u_p^2p & \text{if }\widetilde{E}_d\left(\frac{1-\sqrt{1-\lambda}}{2}\right)\text{ is ordinary at }p\text{ and }\left(\frac{1-\lambda}{p}\right)=1,\\ -\left(\frac{k_d}{p}\right)u_p^2p & \text{if }\widetilde{E}_d\left(\frac{1-\sqrt{1-\lambda}}{2}\right)\text{ is ordinary at }p\text{ and }\left(\frac{1-\lambda}{p}\right)=-1 \end{cases} \pmod{p^3}.$$

The paper states that this conjecture is verified in the cases arising from two displayed examples using Beukers's theorem; the general assertion is therefore not established by the supplied text.