Even-dimensional Dwork hypersurface hypergeometric point count conjecture

Let $d$ be an even integer and let $q equiv 1\pmod d$. The Dwork hypersurface $X_{\lambda}^d$ over $\mathbb F_q$ is given by

$$x_1^d+x_2^d+\cdots+x_d^d=d\lambda x_1x_2\cdots x_d.$$

Even-dimensional Dwork point count conjecture. The number of points over $\mathbb F_q$ on the Dwork hypersurface can be expressed as

$$\frac{q^{d-1}-1}{q-1}$$

plus a sum of Greene's finite field hypergeometric functions. This extends the proved odd-dimensional case, in which the remaining Gauss-sum terms admit such a hypergeometric expression; the even-dimensional case is presented as conjectural here.