Random groups are asymptotically almost surely graphically discrete

Consider random groups in the few-relator model and in Gromov's density model, with the length of the chosen relators tending to infinity. A group is graphically discrete when it has only trivial lattice envelopes.

Random-group graphical discreteness conjecture. Random groups in the few-relator model and Gromov's density model are graphically discrete asymptotically almost surely as the length of the chosen relators tends to infinity.

This predicts graphical discreteness with probability tending to one in both random-group models, but the supplied text gives no resolution or partial result for this assertion.