BCOV's polynomiality conjecture for quintic threefolds

Let $G_g^{\mathrm{BCOV}}$ be the set of genus-$g$ stable graphs for the quintic threefold, with solid, half-dotted-half-solid, and dotted edges. Place the propagators $(T^{\tilde\phi\tilde\phi},T^{\tilde\phi},T)$ on the corresponding edge types and $P_{g,m,n}$ at vertices with $m$ solid and $n$ dotted half-edges. Define

$$f_g^{\mathrm{BCOV}}:=\sum_{\Gamma\in G_g^{\mathrm{BCOV}}}\frac{1}{|\operatorname{Aut}(\Gamma)|}\operatorname{Cont}_\Gamma.$$

Let $X$ be the polynomial variable associated with the quintic mirror coordinate. BCOV's polynomiality conjecture. For $g>1$, $f_g^{\mathrm{BCOV}}$ is a degree $3g-3$ polynomial in $X$. This is the original BCOV Feynman-rule formulation for the genus-$g$ B-model potential of the quintic threefold. The supplied text gives no evidence of resolution.