Rationality and pole-order conjecture for higher-genus BCOV functions

Let $x$ and $y$ be the algebraic coordinates used for the two-parameter model, and let $dis_0$ denote the discriminant component defined by

$$dis_0=(1-432x(1-y))(1-y).$$

For $g\geq 2$, define the propagators by $\mathtt S^{t_1t_1}=\mathtt S^{t_1t_2}=\mathtt S^{t_2t_1}=0$ and

$$\mathtt S^{t_2t_2}=-\frac{1}{\mathtt K_{t_2t_2t_2}}\frac{\partial}{\partial t_2}\log\left(y\frac{\partial t_2}{\partial y}\right),$$

where $\mathtt K_{t_2t_2t_2}:=\partial_{t_2}\partial_{t_2}\partial_{t_2}\mathtt F_0(q,p)$. The dilaton propagators are taken to vanish: $\mathtt S^{t_i\phi}=\mathtt S^{\phi\phi}=0$. Rationality and pole-order conjecture. There exists a rational function $f_g(x,y)$ of the form

$$f_g(x,y)=\frac{\text{polynomial in }x,y}{dis_0^{2g-2}},$$

which reproduces $\mathtt F_g(q,p)$ from the BCOV recursion relation with these propagators.

This asserts a polynomial-over-discriminant description of the higher-genus functions in the chosen coordinates. The source states the claim as an observation for $g\geq2$ and provides no resolution or general proof.