Mallée's parabolic-triangle isobilliard conjecture

Let $B$ be the unit disk in the plane, and for a convex shape $K\subset\mathbb{R}^2$ let $c(K\times B)$ be the length of the shortest closed generalized billiard orbit in $K$. Mallée's isobilliard conjecture. Mallée's parabolic triangle has the smallest area among convex shapes $K\subset\mathbb{R}^2$ satisfying

$$c(K\times B)=1.$$

This is an equivalent formulation of the conjectural optimality of Mallée's example in Wetzel's worm problem. The source says the problem is still open.