Let N = q^k n^2 be an odd perfect number with special prime q satisfying q \equiv k \equiv 1 \pmod 4 and \gcd(q, n)=1.
The recent answer to this MSE question titled "On odd perfect numbers and a GCD - Part V" appears to have successfully completed a proof for the inequation
\gcd(n,\sigma(n^2)) \neq \gcd(n^2,\sigma(n^2)).
If the proof holds water, then this shows that there cannot be an odd perfect number N' = p^j m^2 with special prime p satisfying p \equiv j \equiv 1 \pmod 4 and \gcd(p, m)=1, of the form
N' = \frac{p^j \sigma(p^j)}{2}\cdot{m}.
You may refer to this MSE question (and the answer contained therein) for more information.
