Quantum Dunkl-element isomorphism conjecture for crystallographic Coxeter systems

Let $(W,S)$ be a crystallographic finite Coxeter system, let $qBE(W,S)$ be the associated quantum algebra, and let ${\bf R}[q_s][\tilde{\theta}_s\mid s\in S]$ denote the subalgebra generated by its quantum Dunkl elements. Let $QH^{\ast}(G/B)$ denote the small quantum cohomology ring of the flag variety for the corresponding semisimple Lie group. Quantum Dunkl-element isomorphism conjecture. Theorem 7.2 holds for any crystallographic finite Coxeter system; equivalently, the quantum Dunkl-element subalgebra is canonically isomorphic to the corresponding quantum cohomology ring:

$${\bf R}[q_s][\tilde{\theta}_s\mid s\in S]\simeq QH^{\ast}(G/B).$$

The paper establishes the isomorphism for classical Coxeter groups and for type $G_2$, and proposes its extension to all crystallographic finite Coxeter systems.