In the following paper (http://www.ams.org/journals/jams/1991-04-04/S0894-0347-1991-1119199-X/), C.L. Stewart showed that there exist infinitely many integers $h$ such that the equation

(1) \begin{equation} xy(x+y) = h \end{equation}

has at least 18 solutions in co-prime integers $x$ and $y$. In the same paper, he also conjectured (his conjecture covers far more) that there exists positive numbers $r,c$ such that for all integers $h$ with $|h| \geq r$, the equation (1) has at most $c$ solutions in co-prime integers $x$ and $y$.

My question is, is it reasonable to conjecture that one may take $c = 18$? That is, does there exist infinitely many $h$ for which (1) has at least 19 solutions in co-prime integers $x$ and $y$?

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