I am clearly too late for the party, but I would like to mention my recent works which concern the cohomology and Chow ring of $BPGL_n$ for $n$ not necessarily a prime number:

* Gu, X, _On the cohomology of the classifying spaces of projective unitary groups_, to appear, Journal of Topology and Analysis, doi:[10.1142/S1793525320500211](https://doi.org/10.1142/S1793525320500211), arXiv:[1612.00506](https://arxiv.org/abs/1612.00506),

* Gu, X, _Some torsion classes in the Chow ring and cohomology of $BPGL_n$_, J. London Math Soc, 103(1), 2021, doi:[10.1112/jlms.12368](https://doi.org/10.1112/jlms.12368), arXiv:[1901.10090](https://arxiv.org/abs/1901.10090).