BCOV's polynomiality conjecture with legs

Let $G_{g,n}^{\mathrm{BCOV}}$ be the set of genus-$g$, $n$-leg stable graphs with the BCOV edge types and solid or dotted legs. Define $f_{g,n}^{\mathrm{BCOV}}$ as the sum over these graphs, weighted by automorphism factors and the prescribed leg, edge, and vertex contributions. Let $X$ be the polynomial variable associated with the quintic mirror coordinate. BCOV's polynomiality conjecture with legs. If $2g-2+n>0$, then $f_{g,n}^{\mathrm{BCOV}}$ is a degree $3g-3+n$ polynomial in $X$. This extends the original BCOV Feynman rule to graph amplitudes with legs. The supplied text gives no evidence of resolution.