Fredholm conjecture for the three-body Helmholtz operator on anisotropic Hilbert spaces

Let $P=\Delta+V-\lambda^2$ be the Helmholtz operator on $\mathbb{R}^n$, where $\lambda^2>0$ and $V=\sum_{\alpha\in I}\pi_\alpha^*(V^\alpha)$ is a three-body potential with $V^\alpha\in\langle z^\alpha\rangle^{-2}\mathcal{C}^{\infty}(X^\alpha)$. Fredholm conjecture. There exist anisotropic Hilbert spaces $\mathcal{X}$ and $\mathcal{Y}$ such that

$$P:\mathcal{X}\rightarrow\mathcal{Y}$$

is Fredholm. This is presented as the first step in a series developing Fredholm theory for the three-body problem; the supplied text gives no resolution, so the conjecture remains open.