Hulek–Verrill supercongruence conjecture

For the Hulek–Verrill family $1-tg(\boldsymbol{x})=0$, define

$$g=(1+x_1+x_2+x_3+x_4)(1+1/x_1+1/x_2+1/x_3+1/x_4)$$

and

$$\operatorname{hv}_k=\sum_{m_1+m_2+m_3+m_4+m_5=k}\left(\frac{k!}{m_1!m_2!m_3!m_4!m_5!}\right)^2.$$

Let $a_p$ be the coefficient of $q^p$ in the weight $4$ cusp form of level $14$ beginning $q-2q^2+8q^3+\cdots$. Hulek–Verrill supercongruence conjecture. For every prime $p\ne7$,

$$\sum_{k=0}^{p-1}\operatorname{hv}_k(-1/7)^k\equiv a_p\pmod{p^2}.$$

The conjecture concerns the attractor point $t_0=-1/7$, where the local zeta-function experimentally factors with a weight-$4$ modular form component. No resolution is supplied in the source.