The equivalence of two Pin(2)-equivariant Seiberg-Witten Floer homologies

We show that for a rational homology 3-sphere $Y$ equipped with a self-conjugate spin${}^c$-structure $\mathfrak{s}$, the $\mathrm{Pin}(2)$-equivariant monopole Floer homology of $(Y,\mathfrak{s})$, as defined by Lin, is isomorphic to the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of $(Y,\mathfrak{s})$ defined by Manolescu.