Generic super-rigidity conjecture for index-zero pseudoholomorphic curves

Let $(M,\omega)$ be a closed symplectic manifold of real dimension at least four, and let ${\mathcal J}_{\operatorname{reg}}$ denote a subset of the space of smooth $\omega$-tame almost complex structures. A closed, connected, simple $J$-holomorphic curve has deformation index $0$ when its deformation index is zero, and is super-rigid when the set of all covers of it is an open subset in the moduli space of all $J$-holomorphic curves, so that no sequence of geometrically distinct curves can converge to any of its covers. Generic super-rigidity conjecture. There exists a Baire subset

$${\mathcal J}_{\operatorname{reg}}$$

such that for every $J\in {\mathcal J}_{\operatorname{reg}}$, every closed, connected and simple $J$-holomorphic curve with deformation index $0$ is super-rigid. This conjecture seeks conditions under which generic perturbation of the almost complex structure gives the strongest possible regularity for multiply covered curves, potentially simplifying the definition of Gromov–Witten invariants and clarifying relations between simple curves and their covers. The source does not provide evidence of a resolution.