Allen's Ramanujan conjecture for the graphs $X_p(\delta,a,c)$

Let $p\geq 5$ be an odd prime, let $\delta,a,c\in\mathbb{F}_p^{\times}$, and let $X_p(\delta,a,c)$ be Allen's Cayley graph on the specified subgroup of $\operatorname{GL}(3,p)$. Let $\left(\frac{\cdot}{p}\right)$ denote the Legendre symbol. Allen's conjecture. If

$$\left(\frac{-c}{p}\right)=\left(\frac{a(a-4\delta)}{p}\right)=1,$$

then $X_p(\delta,a,c)$ is Ramanujan. The source gives no resolution of this conjecture.