Caffarelli Perron-solution regularity conjecture in dimension two

Let a strict supersolution be understood in the sense of the Perron process for the one-phase free-boundary problem, and consider solutions obtained as infima of such strict supersolutions. Caffarelli's conjecture. Every solution obtained as the infimum of strict supersolutions is a classical solution in dimension two.

The source explains that Caffarelli proved partial regularity for these Perron solutions, with smooth free boundary outside a set of zero $(n-1)$-dimensional Hausdorff measure. The conjecture proposes full classical regularity in dimension two, motivated by the known entire examples and their behavior at large scale.