Bethe subalgebra description of the full quantum cohomology action for Nakajima quiver varieties

Determine whether, for Nakajima quiver varieties associated to simply-laced Lie algebras 𝔤 of ADE type, the full action algebra 𝔄(q) ⊂ End_{H^•_G(pt)} H^•_G(X) equals the image of a family of Bethe subalgebras of the Yangian 𝒴(𝔤) under the representation 𝒴(𝔤) → End_{H^•_G(pt)} H^•_G(X), for parameters q in T^{reg}.

Background

For Nakajima quiver varieties, equivariant cohomology carries a representation of the Yangian 𝒴(𝔤). The operators generating the commutative subspace Q(q) are identified with trigonometric Casimir Hamiltonians.

The broader, unresolved problem is whether the entire action algebra 𝔄(q) coincides with the image of Bethe subalgebras. This has been proven in some special cases (e.g., type A partial flag varieties), but remains open in general.