Non-degeneracy conjecture for the pseudo-metric on the deformation space

Let $V_{(J,A)}$ be the tangent space described by the relevant system of PDEs, and let $\mathcal{L}_{L^2}$ be the space of $L^2$-regularity tensors representing the infinitesimal deformations, equipped with the pseudo-metric $\mathbf{g}_f$. Define

$$\mathcal{L}_0:=\mathcal{L}_{L^2}\cap\mathcal{L}_{L^2}^{\perp_{\mathbf{g}_f}}$$

and let $\mathfrak J$ denote the canonical symmetry on the ambient space $\mathcal{F}_{L^2}$. Non-degeneracy conjecture. The pseudo-metric $\mathbf{g}_f$ restricted to $\mathcal{L}_{L^2}$ is non-degenerate, equivalently

$$\mathfrak J\big(\mathcal{L}_0\big)=\{0\},$$

or, equivalently, $\mathcal{L}_{L^2}+\mathcal{L}_{L^2}^{\perp_{\mathbf{g}_f}}$ is dense in $\mathcal{F}_{L^2}$. The claim concerns whether the pseudo-Kähler structure yields a non-degenerate metric on the full deformation space; the surrounding discussion explains that the space is described by a system of PDEs and that its elements admit an explicit decomposition involving a vector field and holomorphic quadratic and cubic differentials. No resolution is given in the supplied text.