Asymptotic geometry of non-abelian Hodge theory and Riemann--Hilbert correspondence, rank three ${\tilde{E}}_6$ case

We prove the Geometric P=W conjecture in rank 3 on the three-punctured sphere. We describe the topology at infinity of the related character variety. We use asymptotic abelianization of harmonic bundles away from the ramification divisor and an equivariant approach near the branch points to find the WKB (also known as Liouville–Green or phase-integral) expansion of the involved maps. We analyze the Stokes phenomenon governing their behavior.