Commutativity conjecture for Heisenberg functors on moduli spaces of sheaves

Let $S$ be the smooth projective surface and let $\CM$ denote the relevant smooth moduli space of sheaves. For each $n\in\BN$, write $e_{(0,...,0)}$ for the functor associated with the $n$-tuple of zeroes, and let $D_{\CM}$ be the derived category of $\CM$. Commutativity conjecture. For any $n,m\in\BN$, one has

$$\underbrace{e_{(0,...,0)}}_{n \text{ zeroes}} \circ \underbrace{e_{(0,...,0)}}_{m \text{ zeroes}} \cong \underbrace{e_{(0,...,0)}}_{m \text{ zeroes}} \circ \underbrace{e_{(0,...,0)}}_{n \text{ zeroes}}$$

as functors $D_{\CM}\rightarrow D_{\CM\times S\times S}$. This is proposed as the categorical commutativity relation for the Heisenberg-type operators arising from the zero-degree correspondences; the paper establishes the corresponding elliptic Hall algebra action on $K$-theory only after tensoring with $\BQ$, while this functorial statement is presented as a conjectural strengthening.