Przytycki's phase-transition conjecture for random group actions on CAT(0) cube complexes

Let $S$ be a finite set of generators, let $d\in(0,1)$ be a density, and consider a random group in the Gromov density model with density $d$ and relator length tending to infinity. A group acts without a global fixed point on a finite-dimensional $\mathrm{CAT}(0)$ cube complex if it has no point fixed by the whole group. Przytycki's phase-transition conjecture. Random groups act without a global fixed point on finite-dimensional $\mathrm{CAT}(0)$ cube complexes at densities $d<\frac14$ and have property (T) at densities $d>\frac14$, with overwhelming probability. The conjecture predicts a sharp phase transition at density $\frac14$, complementing the known positive results below $\frac14$ and the known property (T) results above $\frac13$; the intermediate range remains unresolved.