  We extend the basic theory of the coniveau and strong coniveau filtrations to
the $\ell$-adic setting. By adapting the examples of Benoist--Ottem to the
$\ell$-adic context, we show that the two filtrations differ over any
algebraically closed field of characteristic not $2$.
  When the base field $\mathbb{F}$ is finite, we show that the equality of the
two filtrations over the algebraic closure $\overline{\mathbb{F}}$ has some
consequences for algebraic equivalence for codimension-$2$ cycles over
$\mathbb{F}$. As an application, we prove that the third unramified cohomology
group $H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})$ vanishes for a
large class of rationally chain connected threefolds $X$ over $\mathbb{F}$,
confirming a conjecture of Colliot-Th\'el\`ene and Kahn.
