Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity

The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson states that, a smooth Betti moduli space of complex dimension $d$ over a punctured Riemann surface has the dual boundary complex homotopy equivalent to a sphere of dimension $d-1$. Via a microlocal geometric perspective, we verify this conjecture for a class of rank $n$ wild character varieties over the two-sphere with one puncture, associated with any Stokes Legendrian link defined by an $n$-strand positive braid.