Joyce's extended Donaldson–Thomas invariant conjecture

Fix $\mathbb K=\mathbb C$. Let $P$ be a Calabi–Yau $3$-fold, set $\mathcal A=\operatorname{coh}(P)$, and let $\mathfrak F_{\mathcal A}$ and $K(\mathcal A)$ be as in the source. Let $(\tau,T,\leqslant)$ be a permissible stability condition of Gieseker type on $\mathcal A$, and let $C(\mathcal A)$ denote the cone of classes under consideration. Joyce's extended Donaldson–Thomas conjecture. There should exist extended Donaldson–Thomas invariants $\bar{DT}^\alpha(\tau)\in\mathbb Q$ for all $\alpha\in C(\mathcal A)$, unchanged under deformations of $P$ and transforming according to the stated wall-crossing formula under changes of stability condition. Whenever $\operatorname{Obj}_{\rm ss}^\alpha(\tau)=\operatorname{Obj}_{\rm st}^\alpha(\tau)$, so that $DT^\alpha(\tau)$ is defined, one should have

$$\bar{DT}^\alpha(\tau)=DT^\alpha(\tau)\in\mathbb Z.$$

There may also be more complicated systems of rational-valued invariants with analogous deformation-invariance and transformation properties. The conjecture aims to combine the deformation behaviour of Donaldson–Thomas invariants with the wall-crossing behaviour of Joyce's motivic invariants. The author says that proving it is feasible but difficult and would probably require virtual moduli cycle technology; the source gives no resolution.