Transformation formula for logarithmic-type $p$-adic hypergeometric functions

Let $W$ be the coefficient ring, let $\sigma(t)=ct^p$ and $\widehat{\sigma}(t)=c^{-1}t^p$, and let $a\in\mathbb{Z}_p\backslash\mathbb{Z}_{\leq 0}$ satisfy $a^{(r)}=a$ for some $r>0$, where $a^{(r)}$ is the $r$th Dwork prime. Put

$$h(t):=\prod_{i=0}^{r-1}F_{a^{(i)},\cdots,a^{(i)}}(t)_{<p}.$$

Then the transformation formula is

$$\mathscr{F}_{a,\cdots,a}^{\;(\sigma)}(t)=-\widehat{\mathscr{F}}_{a,\cdots,a}^{\;(\widehat{\sigma})}(t^{-1})$$

in the ring $W\langle t,t^{-1},h(t)^{-1}\rangle$, where $\widehat{\mathscr{F}}_{a,\cdots,a}^{\;(\widehat{\sigma})}(t^{-1})$ is defined as $\omega(\widehat{\mathscr{F}}_{a,\cdots,a}^{\;(\widehat{\sigma})}(t))$ and $\mathscr{F}_{a,\cdots,a}^{\;(\sigma)}(t)$ is the $p$-adic hypergeometric function of logarithmic type. This transformation relates the two Frobenius lifts by inversion of the variable; the supplied text gives a proof via comparison of two expressions for $e_{\xi}-\Phi(e_{\xi})$, but does not otherwise establish the status of the displayed formula beyond that proof.