A modulo $p^5$ extension of van Hamme's (A.2) supercongruence

Let $p\geq 5$ be a prime satisfying $p\equiv 3\pmod{4}$, and let $\Gamma_p$ denote the $p$-adic Gamma function. A modulo $p^5$ extension of van Hamme's (A.2) supercongruence.

$$\sum_{k=0}^{\frac{p-1}{2}}(-1)^k(4k+1)\left(\frac{\left(\frac{1}{2}\right)_k}{k!}\right)^5\equiv -\frac{p^3}{16}\Gamma_p\left(\frac{1}{4}\right)^4\pmod{p^5}.$$

This is presented as a numerical conjecture that partially motivates the paper, extending the known congruence in the case $p\equiv 3\pmod{4}$ from modulus $p^3$ to modulus $p^5$.