Bridgeland's conjecture on the derived autoequivalence group of twisted K3 surfaces

Let $(X,\alpha)$ be a projective twisted K3 surface, let ${\rm D}^{\rm b}(X,\alpha)$ be its bounded derived category of twisted coherent sheaves, let ${\mathcal P}_0(X,\alpha)$ be the distinguished component of the relevant space of stability conditions, and let ${\rm O}_+(\widetilde H(X,\alpha,{\mathbb Z}))$ be the orientation-preserving isometry group of the twisted Mukai lattice. Bridgeland's conjecture. There exists a natural short exact sequence

$$0\longrightarrow\pi_1({\mathcal P}_0(X,\alpha))\longrightarrow {\rm Aut}({\rm D}^{\rm b}(X,\alpha))\longrightarrow {\rm O}_+(\widetilde H(X,\alpha,{\mathbb Z}))\longrightarrow1.$$

This conjecture would describe the kernel of the cohomological representation and yield the final form of the derived, twisted Global Torelli Theorem; the source says that the answer seems in reach but does not state that the conjecture has been proved.