S-duality of bow varieties

Let $V_i$ and $V_j$ be finite-dimensional vector spaces, and let $\operatorname{GL}(V_i)$ and $\operatorname{GL}(V_j)$ act on the Hamiltonian spaces $\mathbf M_{\medcirc}(V_i,V_j)$ and $\mathbf M_{\times}(V_i,V_j)$, respectively, as described in the source. Bow-variety S-duality conjecture. The Hamiltonian spaces

$$\operatorname{GL}(V_i)\curvearrowright \mathbf M_{\medcirc}(V_i,V_j) \curvearrowleft\operatorname{GL}(V_j)$$

and

$$\operatorname{GL}(V_i)\curvearrowright \mathbf M_{\times}(V_i,V_j) \curvearrowleft\operatorname{GL}(V_j)$$

are S-dual to each other. This conjecture identifies the two basic types of factors in Cherkis bow varieties under S-duality and is used to relate their Coulomb-branch constructions.