Lusztig duality conjecture for Hecke insertion crystals

A decreasing factorization is mapped under Hecke insertion to a semistandard set-valued tableau with entries in $\{1,\ldots,n\}$. The Lusztig dual crystal conjecture. The Lusztig dual of the crystal structure given by Monical and Searles under Hecke insertion corresponds to changing $i \leftrightarrow n+1-i$ in every box and reversing each row. This would give an explicit description of the dual crystal structure on the tableau side and clarify the relationship between Hecke insertion and crystal operations; the source provides experimental evidence but no resolution.