Conjectural supercongruences for truncated hypergeometric series and the $p$-adic Gamma function

Let $p\ge 5$ be a prime. For a rational number $x$ and a positive integer $n$, write

$${}_3F_2\left[\begin{matrix}a_1,&a_2,&a_3\\[5pt]&b_1,&b_2\end{matrix};z\right]_n=\sum_{k=0}^{n}\frac{(a_1)_k(a_2)_k(a_3)_k}{(b_1)_k(b_2)_k}\frac{z^k}{k!}$$

for the truncated hypergeometric series, where $(a)_k$ is the rising factorial, and let $\Gamma_p$ denote the $p$-adic Gamma function. The three supercongruences. Modulo $p^3$,

$${}_3F_2\left[\begin{matrix}\frac12,&\frac13,&\frac23\\[5pt]&1,&1\end{matrix};1\right]_{p-1}\equiv\begin{cases}(-1)^{\frac{p+1}{2}}\Gamma_p\left(\frac16\right)^2\Gamma_p\left(\frac13\right)^2&\text{if }p\equiv1\pmod 6,\\(-1)^{\frac{p-1}{2}}\frac{p^2}{18}\Gamma_p\left(\frac16\right)^2\Gamma_p\left(\frac13\right)^2&\text{if }p\equiv5\pmod 6,\end{cases}$$ $${}_3F_2\left[\begin{matrix}\frac12,&\frac14,&\frac34\\[5pt]&1,&1\end{matrix};1\right]_{p-1}\equiv\begin{cases}(-1)^{\frac{p+1}{2}}\Gamma_p\left(\frac18\right)^2\Gamma_p\left(\frac38\right)^2&\text{if }p\equiv1,3\pmod 8,\\(-1)^{\frac{p-1}{2}}\frac{3p^2}{64}\Gamma_p\left(\frac18\right)^2\Gamma_p\left(\frac38\right)^2&\text{if }p\equiv5,7\pmod 8,\end{cases}$$

and

$${}_3F_2\left[\begin{matrix}\frac12,&\frac16,&\frac56\\[5pt]&1,&1\end{matrix};1\right]_{p-1}\equiv\begin{cases}-\Gamma_p\left(\frac1{12}\right)^2\Gamma_p\left(\frac5{12}\right)^2&\text{if }p\equiv1\pmod 4,\\-\frac{5p^2}{144}\Gamma_p\left(\frac1{12}\right)^2\Gamma_p\left(\frac5{12}\right)^2&\text{if }p\equiv3\pmod 4.\end{cases}$$

These are numerical conjectures extending known modulo-$p^2$ supercongruences; the stated context identifies the case $a=-\frac12$ as already extended modulo $p^3$ by Long and Ramakrishna, while these three cases remain conjectural in the source.