  Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map
$\overline{\Xi}$ between the algebraic compactification of the Dolbeault moduli
space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve
coming from the $\mathbb{C}^\ast$ action, and the analytic compactification of
Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality
equations on a Riemann surface obtained by adding solutions to the decoupled
equations, known as ``limiting configurations''. This map extends the classical
Kobayashi-Hitchin correspondence. The main result of this paper is that
$\overline{\Xi}$ fails to be continuous at the boundary over a certain subset
of the discriminant locus of the Hitchin fibration. This suggests the
possibility of a third, refined compactification which dominates both.
