Equivariant Suslin's conjecture

Let $X$ be a smooth, quasi-projective real variety, and let $L^{q}H\!\mathbb{R}^{a,b}(X)$ denote equivariant real morphic cohomology. Consider the cycle map

$$L^{q}H\!\mathbb{R}^{a,b}(X)\longrightarrow H^{a,b}(X(\C);\underline{\Z}).$$

Equivariant Suslin's conjecture. This map is an isomorphism for $a\leq 0$ with $a\leq q$ and $b\leq q$, and an injection for $a=1$ with $a\leq q$ and $b\leq q$.

The conjecture is the equivariant analogue of Suslin's morphic-cohomology conjecture; the paper states that the complex, real, and equivariant versions are equivalent. Its resolution status is not specified in the supplied text.