Beilinson's standard conjecture for homologically trivial cycles

Let $X_K$ be the generic fiber of the arithmetic or geometric family, let ${\operatorname{Ch}}^i(X_K)^0$ denote the subgroup of homologically trivial codimension-$i$ cycles, and let ${\mathsf L}$ be the Lefschetz operator induced by the ample line bundle. Beilinson's standard conjecture. For $i\leq (n+1)/2$, there is an isomorphism

$${\mathsf L}^{n+1-2i}:{\operatorname{Ch}}^i(X_K)^0\overset{\sim}{\longrightarrow}{\operatorname{Ch}}^{n+1-i}(X_K)^0,$$

and for $x\in\widehat{\operatorname{Ch}}^i(X_K)^0$ with $x\ne0$ and ${\mathsf L}^{n+2-2i}x=0$, one has

$$(-1)^i(x,{\mathsf L}^{n+1-2i}x)>0.$$

The source identifies this with Beilinson's standard conjecture; the supplied text gives no resolution status.