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 $y$?



Integers $h$ such that $xy(x+y) = h$ has many integer solutions
There seem to be lots of $h$ which have $20$ solutions. Among squarefree $47$-smooth numbers I found 
$$101623830, 363993630, 455885430, 1418488890, 15427730010, 31983962010, 322640788470, 1087394017710, 11406069164490, 304250263527210$$ with $20$ solutions and $903210$ and $23730036330$ with $24$.


EDIT: Oops: my program was counting only solutions with $x > 0$ (but $y$ of any sign). Those numbers $101623830, \ldots, 304250263527210$ actually have $30$ solutions each, and $903210$ and $23730036330$ have $36$.


