Chi-independence of the module structure for Betti stacks

Let $r,d\in\mathbf{Z}$ with $r>0$, and let $\mathscr{A}_{\mathbf{Z}_{\geqslant1}\cdot(r,d)}^{\operatorname{Betti}}$ be the indicated Betti CoHA subalgebra object. Let $\mathbb{H}_{lr}$ be the ring of tautological classes acting on the rank-$lr$ component. Chi-independence conjecture for Betti module structures. For every $l\geqslant1$, the homogeneous component $(\mathscr{A}_{\mathbf{Z}_{\geqslant1}\cdot(r,d)}^{\operatorname{Betti}})_{(lr,ld)}$ is stable under the $\mathbb{H}_{lr}$-action. If $d'\in\mathbf{Z}$ satisfies $\gcd(r,d)=\gcd(r,d')$, then there is an isomorphism

$$\operatorname{H}^*((\mathscr{A}_{\mathbf{Z}_{\geqslant1}\cdot(r,d)}^{\operatorname{Betti}})_{(lr,ld)})\cong\operatorname{H}^*((\mathscr{A}_{\mathbf{Z}_{\geqslant1}\cdot(r,d)}^{\operatorname{Betti}})_{(lr,ld')})$$

of $\mathbb{H}_{lr}$-modules. This is posed as the Betti-side generalization of the module-compatibility proposition known under the coprimality condition.