The orientifold CoHA structure conjecture

Let $Q$ be a $\sigma$-symmetric quiver, let $\widetilde{V}_Q$ be the graded primitive object governing its CoHA, and let $W^{\mathsf{prim}}_{Q,e}$ be the primitive orientifold contribution for $e\in\Lambda_Q^{\sigma,+}$. Let $\mathcal{H}_Q(e)$ denote the subalgebra used when $\mathcal{H}_Q$ is supercommutative.

Orientifold CoHA structure conjecture. The CoHA action map

$$\bigoplus_{e\in\Lambda_Q^{\sigma,+}}\operatorname{Sym}((\widetilde{V}_Q)_{(\mathbb{Z}_2,e)})\boxtimes W^{\mathsf{prim}}_{Q,e}\xrightarrow[]{\star}\mathcal{M}_Q$$

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

This is the stronger structural form of the orientifold freeness prediction, describing the full CoHA action rather than only an abstract family of subalgebras. The supplied text gives no resolution status.