The finite Ruth–Aaron triple conjecture

For a positive integer $n=\prod_{i=1}^k p_i^{a_i}$, define

$$f(n)=\sum_{i=1}^k a_i p_i.$$

A Ruth–Aaron triple is a sequence $n,n+1,n+2$ of consecutive integers with equal values of $f$. Finite Ruth–Aaron triple conjecture. There are only finitely many integers $n$ such that

$$f(n)=f(n+1)=f(n+2).$$

The conjecture is motivated by the reported rarity of Ruth–Aaron triples. The source proves only a conditional boundedness result under a growth assumption and a uniform-distribution assumption, so the unconditional assertion remains open.