The Virasoro constraint for moduli of sheaves

Let $X$ be a smooth projective variety and let $M$ be a moduli space of sheaves on $X$ with virtual fundamental class $[M]^{\mathrm{vir}}$. Let $\mathbb{D}^X$ be the descendent algebra and let $\mathsf{L}_{\mathrm{wt}_0}\colon\mathbb{D}^X\to\mathbb{D}^X_{\mathrm{wt}_0}$ be the weight-zero Virasoro operator. Virasoro constraint for sheaves. For every $D\in\mathbb{D}^X$,

$$\int_{[M]^{\mathrm{vir}}}\mathsf{L}_{\mathrm{wt}_0}(D)=0.$$

The paper proves this conjecture in the stated curve and surface cases under its hypotheses, but the formulation is presented generally as a conjectural constraint.