Must the Spectrum of a Random Schr\"odinger Operator Contain an Interval?

We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables. The proof proceeds by extending a result about the presence of ground states for atypical realizations of the classical Anderson model, which we prove here as well and which appears to be new.