Integrality and pullback conjecture for Swan classes

Let $U$ be a smooth connected scheme of dimension $d$ over a perfect field $F$, and let $\mathcal F$ be a smooth $\overline{\mathbb F}_\ell$-sheaf on $U$. Swan class conjecture. (1) The Swan class

$$\operatorname{Sw}(\mathcal F)\in CH_0(\overline U\setminus U)\otimes_{\mathbb Z}\mathbb Q$$

is in the image of $CH_0(\overline U\setminus U)$. (2) If $f:V\to U$ is a finite étale Galois covering trivializing $\mathcal F$, if $\mathcal C_V^{\mathrm{sm},0}$ is cofinal in $\mathcal C_V$, and if the conjecture concerning the log Lefschetz class holds as in the definition of the integral Swan class, then

$$\operatorname{Sw}_{V/U}(\mathcal F)_{\mathbb Z}\in CH_0(\overline V\setminus V)$$

is in the image of

$$f^*:CH_0(\overline U\setminus U)\to CH_0(\overline V\setminus V).$$

The curve case is established in the preceding lemma, and the text says that the analogous assertion is expected in higher dimension; it also notes that the needed assumptions hold when $\dim U\le 2$.