$P=W$ conjecture in lowest degree for rank $2$ over the $5$-punctured sphere

We use abelianization of Higgs bundles away from the ramification divisor and fiducial solutions to analyze the large scale behaviour of Fenchel–Nielsen co-ordinates on the moduli space of rank $2$ Higgs bundles on the Riemann sphere with $5$ punctures. We solve the related Hitchin WKB problem and prove the lowest degree weighted pieces of the $P=W$ conjecture in this case.