Joyce's generic Kähler perturbation conjecture for special Lagrangian moduli spaces

Let $(M,J,\omega,\Omega)$ be an almost Calabi–Yau $m$-fold, let $X$ be a compact special Lagrangian $m$-fold in $M$ with conical singularities, and let $\mathcal I_{X'}$ and $\mathcal O_{X'}$ be as in Theorem. Write $\check{\omega}$ for a Kähler form in the Kähler class of $\omega$, and let $\check{\mathcal M}_X$ denote the moduli space of compact special Lagrangian $m$-folds $\hat X$ with conical singularities in $(M,J,\check{\omega},\Omega)$ that are isotopic to $X$. Joyce's generic Kähler perturbation conjecture. For a second category subset of such Kähler forms $\check{\omega}$, the space $\check{\mathcal M}_X$ is a manifold of dimension

$$\mathop{\rm \dim} \mathcal I_{X'}-\mathop{\rm \dim} \mathcal O_{X'}.$$

The preceding perturbation results show smoothness near any prescribed compact subset after a suitable perturbation, but do not establish a single perturbation for which the entire moduli space is smooth; the conjecture asserts that this holds for a generic Kähler form in the fixed Kähler class.