Does there exist a $k$ such that every sufficiently large integer can be written in the form $$\prod_{i=1}^k a_i - \sum_{i=1}^k a_i$$ for some integers $a_i\geq 2$?

Does there exist a $k$ such that every sufficiently large integer can be written in the form $$\prod_{i=1}^k a_i - \sum_{i=1}^k a_i$$ for some integers $a_i\geq 2$?

Erdős attributes this question to Schinzel. Eli Seamans has observed that the answer is yes (with $k=2$) for a very simple reason: $n = 2(n+2)-(2+(n+2))$. There may well have been some additional constraint in the problem as Schinzel posed it, but [Er61] does not record what this is.