The eccentric commuting-stack object conjecture

Let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime. Let $\mathcal{H}_{m,n}$ and $\mathcal{H}'_{m,n}$ be the derived objects obtained from the ordinary and eccentric flag commuting stacks, respectively, both viewed in $D^b(\operatorname{Coh}_{\mathbb{C}^*\times\mathbb{C}^*}(\operatorname{Comm}_n))$, and let $q_2$ be the elementary character scaling the second matrix. Eccentric commuting-stack conjecture. There is an isomorphism

$$\mathcal{H}_{m,n}\cong\mathcal{H}'_{m,n}\otimes q_2^{n-1}.$$

The claim predicts that the two geometric constructions agree up to the stated equivariant character twist. No proof or counterexample is supplied in the given material.