Reduced real Lawson homology filtration conjecture

Let $X$ be a smooth real projective variety. Let $RT_tH_n(X)$ denote the topological filtration on $H_n(ReX;\mathbb Z_2)$, given by the image of $\Phi_{t,n}$, and let $RG_tH_n(X)$ denote the geometric filtration. Reduced real Lawson homology filtration conjecture. The two filtrations coincide, namely

$$RT_tH_n(X)=RG_tH_n(X).$$

This is the real analogue proposed after recalling the Friedlander–Mazur conjecture for Lawson homology. The source gives no resolution status for the assertion.