The generalized Lefschetz standard conjecture for homological correspondences

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and for each $r\in\mathbf{Q}_{>0}$ let $\gamma_r$ be the homological correspondence acting on $H^i(X)$ by multiplication by $r^i$. Let $G_r$ be a rational algebraic $n$-cycle on $X\times X$, and let $\|\cdot\|$ be a norm on $\mathsf{N}^n(X\times X)_\mathbf{R}$ and $\deg(\cdot)$ the degree with respect to a fixed ample divisor. Generalized Lefschetz standard conjecture. For every $r\in\mathbf{Q}_{>0}$, $\gamma_r$ is algebraic and represented by $G_r$, with

$$\gamma_r=\operatorname{cl}_{X\times X}(G_r).$$

Moreover, for every dynamical correspondence $f$ of $X$, there is a constant $C>0$, independent of $r$ and $f$, such that

$$\|G_r\circ f\|\leq C\deg(G_r\circ f).$$

This is a quantitative strengthening of a standard-conjecture-type algebraicity assertion; the paper proves it in selected cases such as abelian varieties and Kummer surfaces, but it remains open generally.