Parity-corrected Dwork p-adic hypergeometric inversion conjecture

Let $s\geq 1$ be an integer and let $a_1=\cdots=a_s=a\in\mathbb{Z}_p$. Let $\mathscr{F}^{\rm Dw}_{a}(t)$ denote Dwork's $p$-adic hypergeometric function with all $s$ parameters equal to $a$. Let $l$ be the unique integer in $\{0,1,\ldots,p-1\}$ such that

$$a+l\equiv 0\pmod p.$$

Parity-corrected Dwork inversion conjecture. For an odd prime $p$,

$$\mathscr{F}^{\rm Dw}_{a}(t)=((-1)^s t)^l\mathscr{F}^{\rm Dw}_{a}(t^{-1}),$$

and for $p=2$,

$$\mathscr{F}^{\rm Dw}_{a}(t)=\pm((-1)^s t)^l\mathscr{F}^{\rm Dw}_{a}(t^{-1}).$$

The odd-prime case includes the theorem proved in the paper for $s=1$, while the sign ambiguity is introduced to accommodate the characteristic-two counterexample. The paper proposes this as a replacement sufficient for deriving its main theorem; its general status is not resolved in the supplied text.