The Br_Q-equivariant q-stability and twisted-period isomorphism conjecture

Let $Q$ be a quiver of type ADE, let $\operatorname{QStab}_s\operatorname{\mathcal{D}}_\mathbb{X}(Q)$ be its space of $q$-stability conditions, and assume $\operatorname{Re}(s)\geq 2$. Set

$$\nu=(s-2)/2.$$

Let $\widetilde{\mathcal{M}}_Q$ be the twisted-period space, let $P_\nu\colon\widetilde{\mathcal{M}}_Q\to\mathbb{C}^n$ be its period map, and let $\mathcal{Z}_s\colon\operatorname{QStab}_s\operatorname{\mathcal{D}}_\mathbb{X}(Q)\to\mathbb{C}^n$ be the central-charge map. Br_Q-equivariant q-stability and twisted-period isomorphism conjecture. There is a $\operatorname{Br}_Q$-equivariant isomorphism of complex manifolds $\phi_s$ making the diagram with $P_\nu$ and $\mathcal{Z}_s$ commute. In addition,

$$\phi_s^*\mathcal{E}=E,\qquad \phi_s^*\mathrm{d}=\nabla^{\mathrm{CKZ}}_\nu,$$

for the Euler fields and torsion-free flat connections on the two sides. The conjecture proposes a geometric identification between the stability-condition space and the twisted-period space, including compatibility with the Euler fields and flat connections; the supplied text gives no resolution evidence.