Formal quintic polynomiality conjecture for Birkhoff-factorization coefficients

Let $\widetilde{\mathcal{F}}^{\mathsf{B}}_g$ be the genus-$g$ series associated to the formal quintic, let $T=I^{\mathsf{Q}}_1(q)/I^{\mathsf{Q}}_0(q)$ be the mirror coordinate, and let $\mathds{G}_{\mathsf{Q}}[A_2,A_4,A_6,B_1,\dots,B_4,C_0^{\pm1},C_1^{-1},K_2]$ be the polynomial ring defined in the source. Formal quintic polynomiality conjecture. For $g\geq2$, $\widetilde{\mathcal{F}}^{\mathsf{B}}_g(q)$ belongs to this ring; for $g\geq1$ and $k\geq1$, $\frac{\partial^k\widetilde{\mathcal{F}}^{\mathsf{B}}_g}{\partial T^k}(q)$ belongs to this ring; and $\frac{\partial^k\widetilde{\mathcal{F}}^{\mathsf{B}}_g}{\partial T^k}$ is homogeneous of degree $k$ in $C_1^{-1}$. This conjecture expresses polynomiality of the formal quintic theory and is used to organize its holomorphic anomaly equations; no resolution is supplied in the given text.