Virtual pushforward formula for stable pairs on holomorphic symplectic 4-folds

Let $X$ be a holomorphic symplectic 4-fold and let $\beta\in H_2(X,\mathbb{Z})$ be an irreducible curve class. Let $P_n(X,\beta)$ be the moduli space of stable pairs and $M_n(X,\beta)$ the coarse moduli scheme of one-dimensional stable sheaves $F$ with $[F]=\beta$ and $\chi(F)=n$. Let

$$f:P_n(X,\beta)\to M_n(X,\beta),\qquad (\mathcal{O}_X\to F)\mapsto [F],$$

be the forgetful map. Let $\pi_M:M_n(X,\beta)\times X\to M_n(X,\beta)$ be the projection, and let $\mathbb{F}$ be a universal sheaf, if one exists. Virtual pushforward conjecture. There exists a choice of orientation such that

$$f_*[P_n(X,\beta)]^{\mathrm{vir}}=c_{1-n}\left(-\mathbf{R}\pi_{M*}(\mathbb{F})\right)\cap[M_n(X,\beta)]^{\mathrm{vir}}.$$

This conjectural formula is motivated by the Thom–Porteous formula and relates the stable-pair virtual class to the virtual class of the sheaf moduli space. The supplied text gives no evidence that the formula has been proved or refuted, and the universal sheaf is conditional on existence.