The BS/PT wall-crossing formula with insertions

Let $X$ be a projective threefold with Gorenstein and rational singularities, let $f:Y\to X$ be a resolution of singularities of relative dimension $1$, and let $Y$ be a projective Calabi–Yau threefold. For insertions $\gamma_1,\ldots,\gamma_r\in f^*H^{\cdot}(X,\mathbb{Z})$ and descendant levels $\kappa_1,\ldots,\kappa_r\geqslant 0$, write $\mathrm{BS}^f(q;\gamma,\kappa)$ and $\mathrm{PT}^{Y}(q;\gamma,\kappa)$ for the corresponding generating series, and write $\mathrm{PT}^{Y,\mathrm{exc}}(q)$ for the exceptional PT generating series. BS/PT wall-crossing formula with insertions. The generating series satisfy

$$\mathrm{BS}^f(q;\gamma,\kappa)=\frac{\mathrm{PT}^{Y}(q;\gamma,\kappa)}{\mathrm{PT}^{Y,\mathrm{exc}}(q)}.$$

This extends the Bryan–Steinberg wall-crossing formula from invariants without insertions to arbitrary descendant insertions pulled back from $X$. The source presents the formula as a conjectural statement; its validity concerns the compatibility of the BS/PT wall crossing with descendants in the relative Calabi–Yau resolution setting.