The stability-manifold and autoequivalence conjecture for Kodaira curves

Let $C$ be a reducible singular curve of arithmetic genus one, let $\mathrm{Stab}(C)$ denote its stability manifold, and let $\mathrm{Stab}^{dagger}(C)$ be the connected component containing the distinguished stability conditions. Write $\mathrm{Auteq}(C)$ for the group of exact autoequivalences of $D^b\bigl(\mathrm{Coh}(C)\bigr)$, $\mathrm{Auteq}^0(C)$ for the subgroup acting trivially on the numerical Grothendieck group, and $\mathrm{Pic}^0_{\mathrm{tri}}(C)$ and $\mathrm{Aut}_{\mathrm{tri}}(C)$ for the corresponding triangulated Picard and automorphism groups. Let $\mathrm{P}_0^+(C)$ be the positive degree-zero part of the relevant period domain.

The stability-manifold conjecture. The action of $\mathrm{Auteq}(C)$ on $\mathrm{Stab}(C)$ preserves $\mathrm{Stab}^{dagger}(C)$, and $\mathrm{Stab}^{dagger}(C)$ is simply connected. Consequently, there is an isomorphism

$$\mathrm{Auteq}^0(C)/\bigl(\mathrm{Pic}^0_{\mathrm{tri}}(C)\rtimes\mathrm{Aut}_{\mathrm{tri}}(C)\bigr)\cong\pi_1\mathcal{P}_0^+(C).$$

This is proposed by analogy with the corresponding picture for K3 surfaces and extends the known study of stability manifolds from smooth or irreducible genus-one curves to reducible singular curves. The supplied text does not state a resolution of the conjecture.