Liu's mod-$p^5$ truncated hypergeometric supercongruence

Let $p\geq5$ be a prime with $p\equiv3\pmod{4}$, and let $\Gamma_p$ denote the $p$-adic Gamma function. Liu's conjecture.

$${}_6F_5\bigg[\begin{matrix}\frac{5}{4}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\&\frac{1}{4}&1&1&1&1\end{matrix}\bigg|\ -1\bigg]_{\frac{p-1}{2}}\equiv-\frac{p^3}{16}\Gamma_p\left(\frac{1}{4}\right)^4\pmod{p^5}.$$

The paper's abstract identifies this as a congruence conjectured by J.-C. Liu, and the paper proves it.