Existence and choice-independence conjecture for higher arithmetic degrees

Let $X$ be a smooth projective variety, let $f\colon X\dashrightarrow X$ be a dominant rational map, and let $V\subset X$ be a subvariety of dimension $k$. Let $\lambda_k(f;V)$, $\alpha_k(f)$, and $\alpha_{k+1}(f;V)$ be the arithmetic and dynamical degree quantities defined using an ample divisor $H$, an arithmetically ample model $\overline{\mathcal H}$, and a model $\mathfrak X$. Existence and choice-independence conjecture. The limits defining

$$\lambda_k(f;V),\qquad \alpha_k(f),\qquad \alpha_{k+1}(f;V)$$

exist and are independent of the choices of $H$, $\overline{\mathcal H}$, and the model $\mathfrak X$. The paper explains that these limits are known in some cases, including the relevant first arithmetic degree, but remain unknown in general, especially for $k\geqslant2$.