Polynomial bound for maps between fixed curves

Let $X$ and $Y$ be fixed smooth compact complex curves of genus at least $2$, and let the genus of $X$ be $g$. A polynomial bound conjecture. There is a polynomial function $B(g)$ such that the number of surjective holomorphic maps from $X$ to $Y$ is no more than $B(g)$. This conjecture concerns the dependence on the genus of the source for maps between fixed curves; existing bounds are exponential in $g$, and the true dependence remains open.