Tate's Picard-number conjecture for surfaces over finite fields

Let $k=\mathbb{F}_q$ be a finite field of characteristic $p$, let $X$ be a geometrically connected, smooth, projective $k$-surface, and let $P_2(X,t)$ be the characteristic polynomial of geometric Frobenius on $H^2(\overline{X},\mathbb{Q}_{\ell}(1))$. Let $\rho(X)$ be the Picard number of $X$. Tate's Picard-number conjecture. The number $\rho(X)$ equals the multiplicity of the root $t=q^{-1}$ in $P_2(X,t)$. This is known for numerous classes of surfaces and is equivalent, in the setting described in the source, to the relevant case of the Tate conjecture on algebraic cycles; it remains open in general.