Explicit Littelmann-path formula for duality of homogeneous coordinate rings

Let $M=G/P$ and $M^{opp}=G/P^{opp}=G/Q$ be the flag manifolds equipped with the very ample linearized line bundles whose Mumford quotients are related by the isomorphism induced by the bundle map $\hat{\Theta}$. Let $\widetilde{\Theta}$ denote the induced isomorphism of their homogeneous coordinate rings, and use Littelmann path models for the graded summands of these rings.

Path-reversal conjecture. In these Littelmann path models, $\widetilde{\Theta}$ is given by reversing Lakshmibai–Seshadri paths and translating the initial points of the reversed paths to the origin.

This conjecture proposes an explicit combinatorial description of the duality isomorphism at the level of homogeneous coordinate rings; the source gives no resolution, so whether this path formula indeed computes $\widetilde{\Theta}$ remains open.