Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles

The Hausel-Thaddeus conjectures concern topological mirror symmetry between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles on a curve. A non-archimedean approach was introduced by Groechenig, Wyss and Ziegler, proving the conjecture for coprime rank and degree. This article is concerned with its generalisation to the non-coprime case. We treat both the classical ($D=K$) and meromorphic ($D>K$) settings. We prove an equality of $p$-adic volumes twisted by gerbes between moduli spaces of $SL_n$ and $PGL_n$ Higgs bundles of arbitrary degree. In the meromorphic case, building on results of Maulik and Shen, we show that these twisted $p$-adic volumes are related to intersection cohomology. We also conjecture a connection between these $p$-adic volumes and $BPS$ cohomology.