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.

As mentioned in the question, at density $<1/6$, [Ollivier--Wise][1] showed that a random group is the fundamental group of a compact, non-positively curved cube complex. By [Agol's theorem][2], such groups are virtually special in the sense of [Haglund--Wise][3], 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][4] and [Montee][5] 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.

The bottom line is that, unless a hyperbolic group with (T) happens to be linear, we have no tools to prove residual finiteness. Away from the context of random groups, a very concrete class of examples is provided by the recent [Caprace--Conder--Kaluba--Witzel census][6] of generalised triangle groups. Some of these have since been [shown][7] to be residually finite, again by Ashcroft, but since the methods use cubulation these ones certainly don't have (T).


  [1]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwj1vejzl4L5AhXHh1wKHXhqDAwQFnoECAMQAQ&url=http%3A%2F%2Fwww.yann-ollivier.org%2Frech%2Fpubls%2Fsixthcubes.pdf&usg=AOvVaw2Gzerup-rJoFvtRF0WWI4y
  [2]: https://arxiv.org/abs/1204.2810
  [3]: https://link.springer.com/article/10.1007/s00039-007-0629-4
  [4]: https://arxiv.org/abs/1407.0332
  [5]: https://arxiv.org/abs/2106.14931
  [6]: https://arxiv.org/abs/2011.09276
  [7]: https://arxiv.org/abs/2012.09019