Following my answer to Algebraic Attacks on the Odd Perfect Number Problem, I would like to know whether the argument of quid, namely that if a hypothetic odd perfect number $n$ is such that $\displaystyle{n=\prod_{i}p_{i}^{e_{i}}}$, then $\forall i$ $\overline{\sigma}(p_{i})$, where $\overline{\sigma}$ maps $p_{i}$ to the product of the primes dividing $\sigma(p_{i}^{e_{i}})$, could be the product of several of the $p_{j}$, is compatible with the ABC conjecture. I kind of think there must exist an obstruction to this, but I can't exactly figure out which. Any help would be greatly appreciated.
Thanks in advance.
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