Planar maximizers for area among reduced bodies of fixed minimal width (Euclidean conjecture)

Prove the conjecture that, in the Euclidean plane R^2 and for any fixed minimal width w > 0, the only reduced convex bodies that maximize area are the disk of radius w/2 and the quarter-disk of radius w.

Background

The reverse isominwidth problem asks for the maximal area given fixed minimal width. While a maximizer need not exist among all convex bodies, the problem is meaningful within the class of reduced bodies.

The paper cites a conjecture identifying the precise maximizers in the Euclidean plane; progress has been made for polygons (regular reduced k-gons maximize area), but the full conjecture over all reduced bodies remains open.