  We consider several conjectures on the independence of $\ell$ of the \'etale
cohomology of (singular, open) varieties over $\bar{\mathbf F}_p$. The main
result is that independence of $\ell$ of the Betti numbers
$h^i_{\text{c}}(X,\mathbf Q_\ell)$ for arbitrary varieties is equivalent to
independence of $\ell$ of homological equivalence $\sim_{\text{hom},\ell}$ for
cycles on smooth projective varieties. We give several other equivalent
statements. As a surprising consequence, we prove that independence of $\ell$
of Betti numbers for smooth quasi-projective varieties implies the same result
for arbitrary separated finite type $k$-schemes.
