Teh's filtration conjecture for real varieties

Let $X$ be a smooth projective real variety. For $0\leq q\leq n$, let $RT_qH_n(X)$ denote the image of the generalized cycle map from reduced Lawson homology, and let $N_{2n-q}H_n(X(\mathbb{R});\mathbb{Z}/2)$ be the corresponding niveau filtration step.

Teh's conjecture.

$$RT_qH_n(X)\subseteq N_{2n-q}H_n(X(\mathbb{R});\mathbb{Z}/2)$$

and moreover

$$RT_qH_n(X)=N_{2n-q}H_n(X(\mathbb{R});\mathbb{Z}/2).$$

The conjecture proposes that the topological filtration coming from generalized cycle maps agrees with the niveau filtration on the Borel-Moore homology of real points. The second part is false because the $s$-maps are not always surjective.