Let $X/k$ be a smooth projective geometrically integral variety ($X = A$ an Abelian variety suffices) over $k = \mathbf{F}_q$ with absolute Galois group $\Gamma$, $\bar{X} = X \times_k \bar{k}$, $q = p^n$ and $\ell \neq p$ be a prime.

Is it known if $\dim \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1))^\Gamma = \dim \mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1))^\Gamma$?

Perhaps this follows (for Abelian varieties) from the Tate conjecture in the Hom form $\mathrm{End}(A) \otimes_\mathbf{Z} \mathbf{Q}_\ell = \mathrm{End}_\Gamma(V_\ell A)$ since the left hand side is independent of $\ell$ and this formula holds for $\ell = p$ as well (Zarhin, *Homomorphisms of abelian varieties over finite fields*)?

Edit: What I know:

$\dim \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1)) = \dim \mathrm{H}^2_{crys}(\bar{X}/W) \otimes \mathbf{Q}_p$ (by Katz-Messing since crystalline cohomology is a Weil cohomology theory)

$\mathrm{H}^2_{et}(X,\mathbf{Q}_\ell(1)) = \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1))^\Gamma$ and $\mathrm{H}^2_{fppf}(X,\mathbf{Q}_p(1)) = \mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1))^\Gamma$ (by the Hochschild-Serre spectral sequence since $\mathrm{cd}(\Gamma) = 1$).

$\dim \mathrm{H}^2_{fppf}(X,\mathbf{Q}_\ell(1)) \geq \rho(X) := \mathrm{rk}(\mathrm{NS}(X))$

$\dim \mathrm{H}^2_{et}(X,\mathbf{Q}_\ell(1)) = \rho(X)$ if $X$ is a product of smooth projective curves and Abelian varieties (by Tate,

*Endomorphisms of Abelian Varieties over Finite Fields*, p. 143, Theorem 4)$\dim \mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1)) \leq \dim \mathrm{H}^2_{cris}(\bar{X}/W) \otimes \mathbf{Q}_p$ (by Illusie,

*Complexe de de Rham-Witt et cohomologie crystalline*, p. 627, Théorème 5.5 (5.5.3) or p. 631, Théorème 5.14) (one even has an injection $\mathrm{H}^2_{fppf}(\bar{X},\mathbf{Q}_p(1)) \hookrightarrow \mathrm{H}^2_{cris}(\bar{X}/W) \otimes \mathbf{Q}_p$, I don't know if it is $\Gamma$-equivariant)

(How) can I conclude $\dim \mathrm{H}^2_{fppf}(X,\mathbf{Q}_p(1)) = \rho(X)$ for $X$ a product of smooth projective curves and Abelian varieties from this?

So the question is:

Do we have $\dim \mathrm{H}^2_{et}(\bar{X},\mathbf{Q}_\ell(1))^\Gamma = \dim \mathrm{H}^2_{et}(X,\mathbf{Q}_\ell(1)) = \dim (\mathrm{H}^2_{cris}(\bar{X}/W) \otimes \mathbf{Q}_p)^\Gamma$