Joyce's stability-independent counting invariant conjecture for K3 and abelian surfaces

Let $X$ be a K3 surface or an abelian surface. Let $N(X)$ be the numerical Grothendieck group, let $C(X)\subset N(X)$ be the image of $\operatorname{Coh}(X)\to N(X)$, let $\Lambda$ be the coefficient algebra, and let $\operatorname{Stab}^{\ast}(X)$ be the specified space of Bridgeland stability conditions. For $\sigma\in\operatorname{Stab}^{\ast}(X)$ and $\alpha\in N(X)$, write $J^{\alpha}(\sigma)$ for the weighted counting of $\sigma$-semistable objects of numerical type $\alpha$, and let $\hat{J}^{\alpha}$ denote Joyce's invariant for semistable sheaves.

Joyce's conjecture. For every $\sigma\in\operatorname{Stab}^{\ast}(X)$ and $\alpha\in N(X)$, there is an invariant $J^{\alpha}(\sigma)\in\Lambda$ such that $J^{\alpha}(\sigma)$ does not depend on the choice of $\sigma$; hence it may be denoted by $J^{\alpha}\in\Lambda$. If $\alpha\in C(X)$, then

$$J^{\alpha}=\hat{J}^{\alpha}.$$

The conjecture extends Joyce's counting invariants from semistable sheaves to semistable objects in the derived category and relates them to Bridgeland stability conditions. The cited source presents it as Joyce's proposal; no resolution is supplied here.