This is an open question: there are no densities $1/2>d\geq 1/6$ where a random group is known to be residually finite. Any progress would be a major step forward.
At density <1/6, Ollivier--Wise showed that a random group is the fundamental group of a compact, non-positively curved cube complex. By Agol's theorem, such groups are virtually special in the sense of Haglund--Wise, and in particular residually finite.
At density $\geq 1/2$, random groups are a.a.s. finite.
Ashcroft's paper mentioned in the question, like the previous work of Mackay--Przytycki and Montee improving the Ollivier--Wise bound, gives a non-trivial action on a cube complex (which is enough to contradict (T)), but not the proper, cocompact action needed to apply Agol's theorem.