The coniveau equality for differential forms

Let $F$ be the ambient field over $k$, let $\Omega^j_{F|k}$ be the space of $j$-forms over $k$, and let $N_j$ denote the level filtration defined by subfields of transcendence degree $j$ over $k$. Write $\Omega^j_{F|k,\mathrm{closed}}$ for the closed $j$-forms. Coniveau equality. One has

$$N_j\Omega^j_{F|k}=N_j\Omega^j_{F|k,\mathrm{closed}}=\Omega^j_{F|k,\mathrm{closed}}.$$

The statement relates the level filtration to closed differential forms; the source gives no resolution status beyond presenting it as a conjectural claim.