PROVED (LEAN)
This has been solved in the affirmative and the proof verified in Lean.
Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$?
This would follow immediately from the twin prime conjecture. The answer is yes, proved by Ford, Luca, and Pomerance
[FLP10], who in fact prove there are at least\[\exp((\log\log x)^c)\]many $a\leq x$ such that $a=\phi(n)=\sigma(m)$ for some $n,m$, where $c>0$ is an absolute constant. This lower bound was improved to\[\exp((\log\log x)^{\omega(x)})\]for some $\omega(x)\to \infty$ by Garaev
[Ga11].
This is problem B38 of Guy's collection
[Gu04].
