Equality conjecture for two truncated hypergeometric sums

Let $p$ be an odd prime and let $r$ be a positive integer. Equality conjecture. One should have

$$\sum_{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac12)_k^4}{k!^4} \equiv \sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac12)_k^3}{k!^3}2^{2k} \pmod{p^{r+4}}.$$

The conjecture is proposed because both sides are predicted to have the same Bernoulli-number expansion for $p>3$, and the source notes that it also applies when $p=3$.