The norm comparison conjecture for dynamical correspondences

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$. Let $\|\cdot\|_{\iota}$ denote the chosen operator norm on $\ell$-adic étale cohomology and $\|\cdot\|$ the operator norm on numerical cycle classes. Norm comparison conjecture. For every dynamical correspondence $f$ of $X$, there is a constant $C>0$, independent of $f$, such that

$$\big\|f^*|_{H^{2k}(X)}\big\|_{\iota}\leq C\big\|f^*|_{\mathsf{N}^k(X)_\mathbf{R}}\big\|$$

for every $0\leq k\leq n$, and

$$\big\|f^*|_{H^{2k+1}(X)}\big\|_{\iota}\leq C\sqrt{\big\|f^*|_{\mathsf{N}^k(X)_\mathbf{R}}\big\|\big\|f^*|_{\mathsf{N}^{k+1}(X)_\mathbf{R}}\big\|}$$

for every $0\leq k\leq n-1$. The conjecture is introduced as a quantitative strengthening of the standard conjecture of Künneth type and is claimed to imply the dynamical degree comparison, generalized Weil, and generalized semisimplicity conjectures; it remains open in general.