Andrew Newman and I recently showed that random 2-dimensional hypertrees (Q-acyclic complexes) are aspherical, in [Topology and geometry of random 2-dimensional hypertrees](https://arxiv.org/abs/2004.13572).

This is based, in part, on [earlier work of Costa and Farber][1], who showed that for a wide range of parameter, the Linial–Meshulam random 2-complex (with complete 1-skeleton and independent 2-dimensional faces) is *almost* aspherical, in the sense that if you delete one face from every tetrahedron boundary, you get an aspherical complex.


  [1]: https://arxiv.org/abs/1211.3653