Strong-disorder localisation under absence of supported formal solutions

Let $H(g)$ be a $\Gamma$-trimmed random Schrödinger operator on $\mathbb{Z}^d$ satisfying the Assumptions. Suppose that the complement of $\Gamma$ is a union of finite connected components, and let $\lambda \in \mathbb{R}$ be such that the eigenvalue equation

$$H(0)\psi=\lambda\psi$$

has no non-trivial formal solution $\psi$ supported on $\Gamma^c$. Strong-disorder localisation conjecture. Then the fractional-moment bound holds for sufficiently large $g$.

This would generalise the two preceding localisation theorems: localisation away from the spectrum of the trimmed operator, and localisation under the double-insulation condition. The conjecture concerns Anderson localisation at energies where no formal solution supported on the untrimmed components obstructs the fractional-moment method.