Let for integers $n\geq 1$ the Euler's totient function $\varphi(n)$, and $\sigma(n)$ denotes the sum of divisors $\sum_{d\mid n}d$. I don't know if next question involving the equation $(1)$ was in the literature.
On assumption of the Rassias' conjecture, see this Wikipedia, it's easy to deduce that there exist infinitely many triples $(x,y,z)$ of different positive integers that satisfy $$\varphi(x)(1+\varphi(y))=\sigma(z).\tag{1}$$
Question. Is it possible to deduce, unconditionally, that do exist infinitely many triples $(x,y,z)$ of different integers satisfying $(1)$? If such question is yet known feel free to answer this question as a reference request (thus refer the literature and I try find and read those facts or propositions about this equation). Many thanks.