Equivariant Seiberg–Witten and Heegaard Floer contact invariants for branched covers

Let $K$ be a transverse knot in $S^3$, let $\Sigma_2(K)$ be its double branched cover, and let $\widetilde{\xi}$ be the induced contact structure. Let

$$\widetilde{H}^{\mathbb{Z}_2}_*(SWF(\Sigma_2(K));\mathbb{F}_2)$$

and

$$HF^{\mathbb{Z}_2}_*(\Sigma_2(K);\mathbb{F}_2)$$

be the corresponding equivariant Seiberg–Witten Floer and Heegaard Floer homology groups, viewed as $\mathbb{Z}_2[Q]$-modules, and let $c_2(S^3,\xi_{\operatorname{std}},K)$ and $c_{\mathbb{Z}_2}(\widetilde{\xi})$ be the associated contact invariants. The equivariant contact-invariant comparison conjecture. There is a functorial isomorphism

$$\Psi:\widetilde{H}^{\mathbb{Z}_2}_*(SWF(\Sigma_2(K));\mathbb{F}_2)\longrightarrow HF^{\mathbb{Z}_2}_*(\Sigma_2(K);\mathbb{F}_2)$$

of $\mathbb{Z}_2[Q]$-modules such that

$$\Psi\bigl(c_2(S^3,\xi_{\operatorname{std}},K)\bigr)=c_{\mathbb{Z}_2}(\widetilde{\xi}).$$

This would identify the Seiberg–Witten and Heegaard Floer equivariant contact classes for branched covers; the source gives no resolution evidence.