Manolescu–Lidman-type conjecture for real monopole Floer homology of links

Let $L$ be an oriented link with non-zero determinant. Let $SWF(L)$ be the link Floer stable homotopy object, let $\frak{s}_L$ be the spin structure determined by the orientation of $L$, and let $\widehat{HMR}_*$, % \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{{HMR}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{HMR}{\scalebox{-1}{\tmpbox}}% _*, and $\overline{HMR}_*$ denote the real monopole Floer homologies introduced in the cited work. Write $H^{\mathbb{Z}_2}_*$, $\mathrm{c}H^{\mathbb{Z}_2}$, and $\mathrm{t}H^{\mathbb{Z}_2}$ for equivariant Borel, coBorel, and Tate homologies, respectively. Manolescu–Lidman-type conjecture. There are isomorphisms

$$\widehat{HMR}_*(L,\frak{s}_L;\mathbb{Z}_2)\cong H^{\mathbb{Z}_2}_*(SWF(L);\mathbb{Z}_2),$$ % \savestack{\tmpbox}{\stretchto{% \scaleto{% \scalerel*[\widthof{{HMR}}]{\kern-.6pt\bigwedge\kern-.6pt}% {\rule[-\textheight/2]{1ex}{\textheight}}%WIDTH-LIMITED BIG WEDGE }{\textheight}% }{0.5ex}}% \stackon[1pt]{HMR}{\scalebox{-1}{\tmpbox}}% _*(L,\frak{s}_L;\mathbb{Z}_2)\cong \mathrm{c}H^{\mathbb{Z}_2}_*(SWF(L);\mathbb{Z}_2),

and

$$\overline{HMR}_*(L,\frak{s}_L;\mathbb{Z}_2)\cong \mathrm{t}H^{\mathbb{Z}_2}_*(SWF(L);\mathbb{Z}_2).$$

This conjecture seeks a real analogue of Manolescu–Lidman's isomorphism, identifying the three versions of real monopole Floer homology with the corresponding equivariant homology theories of $SWF(L)$. The supplied text gives no resolution status.