The unfoldability conjecture for billiard triangles in Weyl chambers

Let $\Delta\subset E$ be a Weyl chamber with tip at a special vertex $o$. Let $x_0,x_k\in\Delta$ be special vertices, and consider a billiard triangle with geodesic sides $\overline{ox_0}$ and $\overline{ox_k}$ and broken side $x_0x_1\ldots x_{k-1}x_k$. A billiard triangle is unfoldable when it can be unfolded to a geodesic triangle in the ambient symmetric space or building. It is a generalized Littelmann triangle when its broken side is a weak LS path, namely a broken geodesic satisfying Littelmann's axioms without requiring $\operatorname{dist}(\lambda_{i-1},\lambda_i)=1$ in the definition of an $a$-chain.

Unfoldability conjecture. A billiard triangle in $\Delta$ with sides $\overline{ox_0}$, $\overline{ox_k}$ and $x_0x_1\ldots x_{k-1}x_k$, and with the special vertices $0,x_0,x_k$, is unfoldable if and only if it is a generalized Littelmann triangle.

This conjecture characterizes unfoldable billiard triangles through weak LS paths, extending the known correspondence between Littelmann triangles and tensor-product inclusions for representations of the dual group. The supplied source does not state whether the conjecture has been resolved.