The analytic stability data and nonlinear connection correspondence conjecture

Let $\Gamma$ be a lattice and let $X$ be a connected component of $Stab(Vect_\Gamma)$, with $\phi$ the projection to the space of central charges. Consider the data and axioms listed in items 1)-8), including the nonlinear meromorphic flat connection, its ${\bf C}^\ast$-equivariance, the prescribed lifts of vector fields, and the local conditions at $t=0$.

Analytic stability data correspondence conjecture. The data and axioms described in 1)-8) are in bijection with the set of continuous families of analytic stability data parametrized by $X$.

The claim proposes an equivalence between geometric nonlinear-connection data and continuous families of analytic stability data over a connected component of the stability-data space.