The geometric generation conjecture for commuting stacks

Let $n\in\mathbb{N}$, let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime, and let $d\geq 1$. For the equivariant derived category $D^b(\operatorname{Coh}_{\mathbb{C}^*\times\mathbb{C}^*}(\operatorname{Comm}_n))$, write $\star$ for the $K$-theoretic Hall convolution product and $\mathcal{H}_{m,n}$ for the object associated with the coprime pair $(m,n)$. Geometric generation conjecture. There is an action of $S_d$ on

$$\underbrace{\mathcal{H}_{m,n}\star\dots\star\mathcal{H}_{m,n}}_{d\text{ times}}\in D^b(\operatorname{Coh}_{\mathbb{C}^*\times\mathbb{C}^*}(\operatorname{Comm}_{nd})).$$

If $\bar{\mathcal{E}}_{md,nd}$ denotes its $S_d$-antisymmetric part, then its $K$-theory class is the generator corresponding to $(md,nd)$, and the category $D^b(\operatorname{Coh}_{\mathbb{C}^*\times\mathbb{C}^*}(\operatorname{Comm}_n))$ is generated by

$$\left\{\bar{\mathcal{E}}_{m_1,n_1}\star\dots\star\bar{\mathcal{E}}_{m_k,n_k}\right\}^{\frac{m_1}{n_1}\leq\dots\leq\frac{m_k}{n_k}}_{\substack{(m_i,n_i)\in\mathbb{Z}\times\mathbb{N},\\ n_1+\dots+n_k=n}}.$$

Here generation means that every object is isomorphic to a chain complex built from these objects and their equivariant shifts. This conjecture predicts a geometric lift of the elliptic-Hall-algebra generators and a generating collection for the derived category of the commuting stack; its resolution would describe the object-level behavior of the proposed functor.