The smaller-ring refinement of the perverse stable-pair functional equation

Let $X$ be the Calabi–Yau threefold under consideration, let $\beta\in H_2(X, {\mathbb{Z}})$, and let ${}^p\operatorname{PT}_\beta(e^{iu},Q)$ be the perverse stable-pair generating function. Define

$$R={\mathbb{C}}\left[Q^{\pm 1}, \frac{1}{1-Q}\right][u^{-1},u]].$$

Smaller-ring refinement. The preceding proposition still holds with this smaller ring $R$ in place of the ring allowing all factors $(1-Q^j)^{-1}$; in particular,

$${}^p\operatorname{PT}_\beta(e^{iu},Q)\in R_{\mathsf{w}\cdot\beta},$$

where $R_a=\\{f\in R:f(Q,u)=Q^a f(Q^{-1},-u)\\}$.

This asserts a stronger form of the proposition's symmetry statement, namely that only the factor $(1-Q)^{-1}$ is needed in the coefficient ring. The supplied text does not state whether this refinement is proved or remains open, so its status is recorded as open.