Conservativity conjecture for transcendental motives of surfaces

Let $k$ be an algebraically closed field of characteristic $p\geq 0$, let $X$ be a smooth projective surface over $k$, and let $t_{2}(X)$ denote its transcendental Chow motive with rational coefficients. For a prime number $l\neq p$, its étale realization is $H^{*}_{\operatorname{\acute{e}t}}(t_{2}(X))$. Conservativity conjecture. If

$$H^{*}_{\operatorname{\acute{e}t}}(t_{2}(X))=0,$$

then

$$t_{2}(X)=0.$$

This conjecture asserts that the étale realization detects whether the transcendental motive vanishes; its resolution is not specified in the supplied text.