Truong's equality conjecture for cohomological and numerical dynamical degrees

Let $X$ be a smooth projective variety defined over an algebraically closed field $\mathbf{k}$, and let $f\colon X \vdash X$ be a dominant self-correspondence. For a prime $\ell$ different from the characteristic of $\mathbf{k}$, define the cohomological dynamical degree $\chi_i(f)$ by

$$\chi_i(f) \coloneqq \limsup_{m\to\infty}\left\|(f^m)^*\big|_{H^i_{\operatorname{\acute et}}(X,\mathbf{Q}_\ell)}\right\|^{1/m},$$

and define the numerical dynamical degree $\lambda_k(f)$ by

$$\lambda_k(f) \coloneqq \limsup_{m\to\infty}\left\|(f^m)^*\big|_{N^k(X)_\mathbf{R}}\right\|^{1/m},$$

where $N^k(X)_\mathbf{R}$ is the real vector space of codimension-$k$ algebraic cycles modulo numerical equivalence. Truong's conjecture. For every $1\leq k\leq\dim X$, one has

$$\chi_{2k}(f)=\lambda_k(f).$$

Over $\mathbf{C}$ this equality follows from comparison theorems and Hodge theory, while in arbitrary characteristic it is known in some cases, including automorphisms of smooth projective surfaces. The general statement remains open.