Strong rationality conjecture for cohomological BPS invariants

Let $Q$ be the quiver associated with a noncommutative crepant resolution $A=\mathbb{C}\{Q,W\}$ of the threefold setup, let $C$ be the flopping curve, and let $\Phi(0,1)$ be its dimension vector. Let $\mathcal{H}_{Q,W,\gamma}$ be the cohomological Hall algebra component of dimension vector $\gamma$, let $\operatorname{BPS}_{Q,W,\gamma}$ be the corresponding cohomological BPS invariant, and let $u1_C\in\mathcal{H}_{Q,W,\Phi(0,1)}$ be the degree-zero class defined from the generator $u$ of $\operatorname{H}(\operatorname{pt}/\mathbb{C}^*)$. Strong rationality conjecture. For every $\gamma\in\mathbb{N}^{Q_0}$, the commutator map

$$[u1_C,-]\colon\mathcal{H}_{Q,W,\gamma}\longrightarrow\mathcal{H}_{Q,W,\gamma+\Phi(0,1)}$$

maps $\operatorname{BPS}_{Q,W,\gamma}$ isomorphically to $\operatorname{BPS}_{Q,W,\gamma+\Phi(0,1)}$. The main evidence given is an analogous isomorphism for threefolds of the form $\tilde{X}\times\mathbb{C}$ associated with resolutions of du Val singularities. An implication is that $\operatorname{GV}_{C,r,n}$ is independent of $n$.