Current status of independence of Betti numbers for different Weil cohomology theories

For smooth projective varieties over $k$ separably closed field this should be known: in characteristic zero due to comparisons with Betti-de Rham cohomology, in characteristic $p>0$ due to the Weil Conjectures, esp. Deligne’s work on the Riemann Hypothesis.

For instance, over finite fields this is in Katz-Messing (see here) and for a smooth projective variety $X$ over a separably closed field $K$ of characteristic $p>0$, you can always find another smooth projective variety $X’$, this time defined over the separable closure of a finite field, such that $X’$ is the closed geometric fiber of a smooth relatively projective family $\mathcal{X}\to S$ whose geometric generic fiber is $X$.

If your cohomology is reasonable it should satisfy a version of proper base change, and so the Betti numbers of $X$ and $X’$ are the same and you apply the Katz-Messing result to $X’$.

Likewise if $X$ is defined over a separably closed field of characteristic zero, one can spread it out and specialize it to a smooth projective variety $X’$ defined over the separable closure of a number field, and if your cohomology satisfies proper base change, then the Betti numbers of $X$ are the same as the Betti numbers of (any such) $X’$.

As another instance, for $X’$ over the separable closure of a number field $K$, you can spread it out to some smooth projective family $\mathcal{X}\to V$ where $V$ is open in $\text{Spec}(R)$, $R$ a Dedekind ring, and the algebraic de Rham cohomology of $\mathcal{X}$ over $R$ interpolates both the algebraic de Rham cohomology of $X’/K$ and the crystalline cohomology of every closed fiber of $\mathcal{X}$, so all Betti numbers agree. The algebraic de Rham cohomology of $X’/K$ (base changed to the complex numbers along some complex embedding of $K$ into $\mathbf{C}$) further compares to the holomorphic de Rham cohomology of $X’(\mathbf{C})$ (a theorem of Grothendieck: see here) which further compares to Betti cohomology with $\mathbf{C}$ coefficients, which in turn compares with $\overline{\mathbf{Q}}_{\ell}$-étale cohomology (a consequence of a Theorem of Artin: see SGA 4 Exp. XI and bootstrap from the case of finite coefficients using the fact that $X’(\mathbf{C})$ has a basis of opens trivializing cohomology with constant coefficients, together with the fact that the $H^*(X’(\mathbf{C}),\mathbf{Z}/\ell^n)$ satisfy the Mittag-Leffler condition, so you can bring the inverse limit in $n$ inside cohomology).

All the respective Betti numbers agree, therefore.

If you refer to the compactly supported $\ell$-adic cohomology of arbitrary varieties (i.e. beyond the smooth case, but it seems you’re asking about smooth projective varieties), its dimensions are not known to be independent of $\ell$. This would follow from a combination of the standard conjectures (see here).