No Dris-index solution for the Euler equation with special exponent $k = 5$ and $s \ge 2$

Let $p$ be an odd prime with $p \equiv 1 \pmod 4$, let $m$ be odd with $p \nmid m$, and let $s \ge 2$ be a natural number. The Euler equation for an odd perfect number in Euler form with special exponent $k = 5$ has no Dris-index solution: the two relations $2m^2 = \sigma(p^5)s$ and $\sigma(m^2) = p^5s$ cannot both hold. This is the nontrivial $s \ge 2$ case left after the case $s = 1$ is ruled out by the proved lemma $\sigma(p^5) \ne 2m^2$ since $6 \mid 5 + 1$.