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. https://arxiv.org/abs/1612.00506
Gu, X, Some torsion classes in the Chow ring and cohomology of $BPGL_n$. https://arxiv.org/abs/1612.00506
Gu, X, On the cohomology of the classifying spaces of projective unitary groups, to appear, Journal of Topology and Analysis, doi:10.1142/S1793525320500211, arXiv: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, arXiv:1901.10090.