The two-dimensional localisation and higher-dimensional delocalisation conjecture for the trimmed Anderson model

Let $H(g)$ be the trimmed Anderson random Schrödinger operator in dimension $d$, let $H_\Gamma$ be the associated periodic operator, and let $I$ be an interval contained in the absolutely continuous spectrum of $H_\Gamma$. Trimmed Anderson localisation–delocalisation conjecture. For strong disorder $g\gg1$, if $d=2$, then $H(g)$ exhibits localisation in the sense of; if $d\geq3$, then $H(g)$ has absolutely continuous spectrum on $I$. The claim concerns the third possibility in the invariant setting, where $^c$ has an infinite connected component and $I$ lies in a band of absolutely continuous spectrum of $H_\Gamma$. The source presents this as a conjecture and gives no resolution.