The QALE Laplacian index-set and Fredholm conjecture

Let $(X,J,g)$ be a QALE Kähler manifold asymptotic to $\mathbb{C}^m/G$, and let $n$ be the complex codimension of the singular set of $\mathbb{C}^m/G$. The set $\mathcal{I}_X$ consists of the weight pairs satisfying

$$\mathcal{I}_X=\bigl\{(\beta,\gamma)\in\mathbb{R}^2:\beta<0,\quad 2-2n<\gamma<0,\quad\beta+\gamma>2-2m\bigr\}.$$

For generic $\beta,\gamma\in\mathbb{R}$, consider the weighted Laplacian

$$\Delta:C^{k+2,\alpha}_{\beta,\gamma}(X)\longrightarrow C^{k,\alpha}_{\beta,\gamma-2}(X).$$

The QALE Laplacian index-set and Fredholm conjecture. The displayed description of $\mathcal{I}_X$ holds, and the displayed map is Fredholm, with finite-dimensional kernel and cokernel, for generic $\beta,\gamma\in\mathbb{R}$.

These assertions describe the expected exceptional weight region and Fredholm behavior for the Laplacian on QALE Kähler manifolds. The source presents them as beliefs rather than established results, and supplies no resolution evidence.