Friedlander–Mazur conjecture on the topological, geometric and Hodge filtrations

Let $X$ be a smooth projective variety, and let $H_k(X,\mathbb{Q})$ denote its singular homology. The topological filtration $T_pH_k(X,\mathbb{Q})$ and geometric filtration $G_pH_k(X,\mathbb{Q})$ are as defined in the source, and let $\tilde{F}_pH_k(X,\mathbb{Q})$ be the maximal sub-mixed-Hodge structure of span $k-2p$.

Friedlander–Mazur conjecture. For any smooth projective variety $X$ and $k\geq 2p\geq 0$, one has

$$T_pH_k(X,\mathbb{Q})=G_pH_k(X,\mathbb{Q})=\tilde{F}_pH_k(X,\mathbb{Q}).$$

The conjecture relates Lawson homology and algebraic cycle theory by asserting that the topological, geometric and Hodge filtrations coincide. The source states the conjecture but provides no resolution.