The orientifold CoHA freeness conjecture

Let $Q$ be a $\sigma$-symmetric quiver, meaning that it is symmetric and satisfies $\sigma^*\mathcal{E}=\mathcal{E}$. Let $\mathcal{H}_Q$ be its cohomological Hall algebra, $\mathcal{M}_Q$ its orientifold module, and $W^{\mathsf{prim}}_{Q,e}$ the primitive part in dimension vector $e\in\Lambda_Q^{\sigma,+}$. For each $e$, let $\mathcal{H}_Q(e)\subset\mathcal{H}_Q$ be the explicitly defined $\Lambda_Q^{\sigma,+}\times\mathbb{Z}$-graded subalgebra.

Orientifold CoHA freeness conjecture. If $\mathcal{H}_Q$ is supercommutative, then the CoHA action map

$$\bigoplus_{e\in\Lambda_Q^{\sigma,+}}\mathcal{H}_Q(e)\boxtimes W^{\mathsf{prim}}_{Q,e}\rightarrow\mathcal{M}_Q$$

is an isomorphism in $D^{lb}(\mathsf{Vect}_{\mathbb{Z}})_{\Lambda_Q^{\sigma,+}}$, and its restriction to $\mathcal{H}_Q(e)\boxtimes W^{\mathsf{prim}}_{Q,e}$ is a $\mathcal{H}_Q(e)$-module isomorphism onto its image.

This predicts a free-module decomposition of the orientifold CoHA module and yields a factorization of its Grothendieck-group class. The supplied text gives no resolution status.