Here is the Scribd link. (This same document has likewise been uploaded to Academia.)
Update (February 17, 2022 - 9:39 AM [Manila time]): This article has just been announced on math.NT in arXiv.
Here is the abstract:
Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs
$$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$
$$H = \gcd\bigg(n^2,\sigma(n^2)\bigg)$$
and
$$I = \gcd\bigg(n,\sigma(n^2)\bigg).$$
We prove that $G \times H = I^2$. (Note that it is trivial to show that $G \mid I$ and $I \mid H$ both hold.) We then compute expressions for $G, H,$ and $I$ in terms of $\sigma(q^k)/2, n,$ and $\gcd\bigg(\sigma(q^k)/2,n\bigg)$. Afterwards, we prove that if $G = H = I$, then $\sigma(q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set
$$\mathscr{A} = \{m : \gcd(m,\sigma(m^2))=\gcd(m^2,\sigma(m^2))\}$$
has asymptotic density zero.
