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Claim. A note posted under the forum name RomanLeLan on 21 October 2025, linked as a PDF from the site's discussion thread for Problem 250, claims that $A(q)=\sum_{n\ge1}\sigma(n)q^n$ is transcendental for every algebraic $q$ with $0<|q|<1$; at $q=1/2$ this would answer the question yes. The post itself allows that the argument may contain a mistake. The linked file is no longer available (the link returns 404).

Depends on. Nothing in this wiki.

Standing. Rejected. The site's curator, Thomas Bloom, replied the same day that the argument shows only that $A(q)-4A(q^4)$ is transcendental. That implies at most that one of $A(q)$ and $A(q^4)$ is transcendental and says nothing about $A(q)$ itself, so the note does not prove its claim. The problem's answer rests on the refereed proofs of Duverney and Nesterenko.