Long's hypergeometric supercongruence conjecture

Let $(r_1,r_2,q)$ range over

$$\left\{\left(\frac{1}{2},\frac{1}{2},\frac{4}{3}\right),\left(\frac{1}{2},\frac{1}{2},\frac{7}{6}\right),\left(\frac{1}{2},\frac{1}{3},\frac{7}{6}\right),\left(\frac{1}{2},\frac{1}{3},\frac{5}{4}\right),\left(\frac{1}{2},\frac{1}{4},\frac{7}{6}\right),\left(\frac{1}{2},\frac{1}{2},\frac{5}{4}\right)\right\}.$$

Set $\mathbf{\alpha}_{(r_1,r_2)}=\{r_1,1-r_1,r_2,1-r_2\}$ and $\mathbf{\beta}_q=\{1,1,q,2-q\}$. Let $\mathrm{HD}_{(r_1,r_2,q)}$ denote the hypergeometric datum $(\mathbf{\alpha}_{(r_1,r_2)},\mathbf{\beta}_q,1)$. Long's conjecture. For each of these hypergeometric data, there exists an explicit weight $4$ modular form $f_{(r_1,r_2,q)}$ and Dirichlet character $\chi_{(r_1,r_2,q)}$ such that, for all primes $p\geq 7$,

$$p\cdot {}_4F_3\left[\begin{matrix}r_1&1-r_1&r_2&1-r_2\\&1&q&2-q\end{matrix};1\right]_{p-1}\equiv \chi_{(r_1,r_2,q)}(p)a_p(f_{(r_1,r_2,q)})\pmod{p^3}.$$

The conjecture connects these hypergeometric supercongruences with modular forms. Modularity is known for the first three listed hypergeometric data, while the stated congruences and the required generalizations of Dwork's results remain to be established in full.