Birkhoff-factorization coefficient ring conjecture

Let $n\geq0$, let $I_n$ be the equivariant $I$-function, and write its restriction to $H=\lambda_i$ as

$$I_n|_{H=\lambda_i}=e^{\mu/z}\left(R_{0,i}+R_{1,i}z+R_{2,i}z^2+\dots\right),$$

where $R_{k,i}$ are formal series and $\mathds{G}_n$ is the ring defined from the roots $L_i$ and the polynomial $f_n$. Birkhoff-factorization coefficient ring conjecture. For all $k\geq0$,

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

The supplied text says that this conjecture was proved under the specialization referred to as $(\mathrm{sp})$ in Theorem 4 of the cited work, so its database status is solved in that specialization.