The polynomial-generation conjecture for almost $A$-quasi-modular forms

Let $\mathbb{T}_{>0}$ be the algebra of power series with positive convergence radius, and let $g$, $h$, and $\boldsymbol{s}_2$ be the almost $A$-quasi-modular forms used in the source. Let $\mathcal{M}$ denote the algebra introduced above. Polynomial-generation conjecture. One has

$$\mathcal{M}=\mathbb{T}_{>0}[g,h,\boldsymbol{s}_2].$$

This conjecture gives the expected exact image of $\mathcal{M}$ in the ambient algebra involving $g$, $h$, $\boldsymbol{s}_2$, and its inverse. The source presents it as an open conjecture motivated by the expected non-regularity of $\boldsymbol{s}_2^{-1}$.