Lusztig-involution conjecture for general Lascoux polynomials and atoms

Let $\lambda$ be a partition and let $\operatorname{SV}^n(\lambda)$ be the set of semistandard set-valued tableaux of shape $\lambda$. For a tableau $T$, define its right K-key tableau by

$$K(T):=k(\min(T^*)^*),$$

where $T^*$ is obtained using the Lusztig involution on each irreducible crystal component, $\min(T)$ takes the least entry in each box, and $k$ is the right key tableau of a semistandard tableau. Set

$$\operatorname{SV}^n_w(\lambda)=\{T\in\operatorname{SV}^n(\lambda)\mid K(T)\preceq K_{w\lambda}\},$$ $$\overline{\operatorname{SV}}^n_w(\lambda)=\{T\in\operatorname{SV}^n(\lambda)\mid K(T)=K_{w\lambda}\}.$$

Lusztig-involution conjecture. For every partition $\lambda$,

$$L_{w\lambda}(\mathbf{x};\beta)=\sum_{T\in\operatorname{SV}^n_w(\lambda)}\operatorname{wt}_{\beta}(T),$$

and

$$\overline{L}_{w\lambda}(\mathbf{x};\beta)=\sum_{T\in\overline{\operatorname{SV}}^n_w(\lambda)}\operatorname{wt}_{\beta}(T).$$

The preceding key-tableau formulas are proved for rectangular shapes but fail for general partitions; the conjecture proposes that inserting the Lusztig involution repairs the formulas and gives combinatorial interpretations of general Lascoux polynomials and atoms.