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This answer to an earlier (and related) MSE question summarizes what appears to be the first documented "evidence" against the Descartes-Frenicle-Sorli conjecture that $k=1$, if $q^k n^2$ is an odd perfect number with special / Euler prime $q$ (satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$). (Specifically, via exhaustion of all possible cases, it appears plausible that $k>1$ must hold true.)

On the other hand, for the Descartes spoof (which is probably the closest we will ever get to an "actual odd perfect number") $$\mathscr{D} = {3^2}\cdot{7^2}\cdot{{11}^2}\cdot{{13}^2}\cdot{22021} = 198585576189$$ the quasi-Euler prime $22021$ (which is actually composite) has exponent $1$. Additionally, the non-Euler part ${3^2}\cdot{7^2}\cdot{{11}^2}\cdot{{13}^2}$ of $\mathscr{D}$ can be (directly) shown to be deficient-perfect, a condition which is known to be equivalent to the Descartes-Frenicle-Sorli conjecture.

So here is my question:

What other evidence or heuristic do you know of that supports (or otherwise does not support) the Descartes-Frenicle-Sorli conjecture for odd perfect numbers?

One piece of evidence/heuristic per answer only, please.

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