Transformation formula between logarithmic and Dwork p-adic hypergeometric functions

Let $\sigma(t)=ct^p$ and $\widehat{\sigma}(t)=c^{-1}t^p$. Let $a\in\mathbb{Z}_p\backslash\mathbb{Z}_{\leq 0}$, and suppose that the $r$th Dwork prime satisfies $a^{(r)}=a$ for some $r>0$. Define

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

Here $W\langle t,t^{-1},h(t)^{-1}\rangle$ is the indicated ring, and $\omega$ is the map on this ring defined in the cited reference. Transformation formula between logarithmic and Dwork p-adic hypergeometric functions. In $W\langle t,t^{-1},h(t)^{-1}\rangle$,

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

where $\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t^{-1})$ means $\omega(\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t))$. The function $\mathscr{F}_{a}^{\;(\sigma)}(t)$ is the $p$-adic hypergeometric function of logarithmic type. This transformation is known for $s=2$, $a\in\frac{1}{N}\mathbb{Z}$ with $0<a<1$ and $p>N$, but the general claim remains open.