Conjecture on the Weyl group optimization step in GIT algorithms

The algorithms use a Weyl group $W$ and a subset $W'\subset W$ in their final optimization step to remove nonmaximal states. Weyl group optimization conjecture. At the last step of the algorithms, the optimization routine using $W'\subset W$ does not reduce the output of algorithms

$$and$$

. This conjecture is motivated by the authors' computational experiments, in which the final optimization step has not changed the output; its general validity remains open.