Evans's conjectural hypergeometric relation for weight-two newforms

Let $a_3(n)$ and $a_4(n)$ be the Fourier coefficients of newforms

$$\sum_{n\geq1}a_3(n)q^n\in S_2\left(\Gamma_0(972)\right),\qquad \sum_{n\geq1}a_4(n)q^n\in S_2\left(\Gamma_0(768)\right),$$

whose coefficient fields are $\mathbb{Q}\left(\sqrt{2}\right)$ and $\mathbb{Q}\left(\sqrt{3}\right)$, respectively. Let $\chi_6$ and $\chi_8$ denote the relevant characters of orders $6$ and $8$, let $\varphi$ be the quadratic character, let $J$ denote a Jacobi sum, and let ${}_3F_2(\cdots)_q$ denote the finite-field hypergeometric value. Evans's conjecture. If $q\equiv1\pmod{6}$, then

$$\overline{\chi_6}(12)J\left(\chi_6,\chi_6\right)-\overline{\chi_6}(3)J\left(\chi_6^2,\chi_6^2\right)\cdot {}_3F_2\left(\begin{array}{ccc}\overline{\chi_6},&\varphi,&\chi_6\\&\varphi\chi_6,&\varphi\chi_6\end{array}\Big|\frac14\right)_q=\begin{cases}a_3(p),&q=p,\ p\equiv1\pmod{6},\a_3(p)^2+2p,&q=p^2,\ p\equiv5\pmod{6},\end{cases}$$

and if $q\equiv1\pmod{8}$, then

$$\chi_8(-4)J\left(\chi_8,\chi_8\right)-\chi_8(-4)J\left(\chi_8^2,\chi_8^3\right)\cdot {}_3F_2\left(\begin{array}{ccc}\overline{\chi_8},&\chi_8^3,&\chi_8\\&\overline{\chi_8}^2,&\varphi\chi_8\end{array}\Big|\frac14\right)_q=\begin{cases}a_4(p),&q=p,\ p\equiv1\pmod{8},\a_4(p)^2+2p^2,&q=p^2,\ p\not\equiv1\pmod{8}.\end{cases}$$

These are conjectural relations connecting Fourier coefficients of weight-two newforms with non-integral coefficient fields to finite-field hypergeometric functions; the supplied text gives no resolution status.