The square-root abundancy conjecture for spoof odd perfect numbers

Let $n=km>1$ be a spoof odd perfect number. If $\gcd(m,k)=1$, then $\sqrt{k}$ is an integer and the sum-of-divisors function $\sigma$ is evaluated at this factor. Square-root abundancy conjecture. The inequality

$$\sigma(\sqrt{k})<m$$

holds. The claim is consistent with the numerical values in Descartes' example, where $\sigma(\sqrt{k})=5376<22021=m$. The source does not establish it in general.