Parametric supercongruence for a truncated hypergeometric series

Let $p\ge 5$ be a prime. For a $p$-adic integer $a$, let $\langle a\rangle_p$ denote its least nonnegative residue modulo $p$, and let $\Gamma_p$ denote the $p$-adic Gamma function. Parametric supercongruence. If $\langle a\rangle_p\equiv0\pmod 2$, then

$${}_3F_2\left[\begin{matrix}\frac12,&-a,&a+1\\[5pt]&1,&1\end{matrix};1\right]_{p-1}\equiv(-1)^{\frac{p+1}{2}}\Gamma_p\left(-\frac a2\right)^2\Gamma_p\left(\frac{a+1}{2}\right)^2\pmod{p^3}.$$

The source presents this as a further conjectural modulo-$p^3$ extension, supported by strong numerical evidence, so its general validity remains open.