Goodson's point-counting conjecture for Dwork hypersurfaces

Let $d$ be an odd prime, let $p$ be a prime such that $p\not\equiv 1 \pmod{d}$, and let

$$X_{\lambda}^d:\ x_1^d+x_2^d+\cdots+x_d^d=d\lambda x_1x_2\cdots x_d$$

be the Dwork hypersurface over $\mathbb{F}_p$. Write $\,_{d-1}G_{d-1}\!\left[\begin{array}{cccc} \frac{1}{d}, & \frac{2}{d}, & \ldots, & \frac{d-1}{d} \\ 0, & 0, & \ldots, & 0 \end{array}\middle|\lambda^d\right]_p$ for McCarthy's $p$-adic hypergeometric series. Goodson's point-counting conjecture. The number of points of $X_{\lambda}^d$ over $\mathbb{F}_p$ is

$$\#X_{\lambda}^d(\mathbb{F}_p)=\frac{p^{d-1}-1}{p-1}+\frac{1}{p-1}+\,_{d-1}G_{d-1}\!\left[\begin{array}{cccc} \frac{1}{d}, & \frac{2}{d}, & \ldots, & \frac{d-1}{d} \\ 0, & 0, & \ldots, & 0 \end{array}\middle|\lambda^d\right]_p.$$

This conjecture extends the known point-counting formula for Dwork hypersurfaces to primes $p\not\equiv 1\pmod d$, where the usual finite-field hypergeometric expression is unavailable; its resolution is not established in the supplied source.