The moduli conjecture for weakly asymptotically conical special Lagrangian folds

Let $L$ be a weakly asymptotically conical special Lagrangian $m$-fold in $\mathbb C^m$ with cone $C$, and let $k$ be the number of ends of $C$ at infinity. Let $\mathcal M^w_L$ be the moduli space of weakly asymptotically conical special Lagrangian $m$-folds in $\mathbb C^m$ with cone $C$. Let $\operatorname{l-ind}(C)$ be the Legendrian index of $C$, namely the number of eigenvalues of the Laplacian on the link of $C$ in $(0,2m)$, counted with multiplicity. Weakly asymptotically conical moduli conjecture. Near $L$, the space $\mathcal M^w_L$ is a smooth manifold of dimension $b^1(L)+k-1+\operatorname{l-ind}(C)$. The result is stated as an open expected analogue for weakly asymptotically conical folds.