Equivariant Birkhoff-factorization coefficient conjecture for local \u03b2c0^2

Let $R_{k,i}$ be the coefficients in the asymptotic expansion of the restriction of the equivariant $I$-function to $H=\lambda_i$ for local $\mathbb{P}^2$, and let $\mathds{G}_2$ be the coefficient ring generated by the corresponding roots and polynomial factors. Local $\mathbb{P}^2$ coefficient-ring conjecture. For all $k\geq0$,

$$R_{k,i}\in\mathds{G}_2.$$

This is presented as an expected constraint following the normalization used in the cited work; the given text does not state a general proof.