Kato's rank-one Swan class comparison and existence conjecture

Let $X$ be a separated scheme of finite type over a perfect field $F$, let $U\subset X$ be dense, connected, smooth, and purely $d$-dimensional, let $\ell$ be invertible in $F$, and let $\mathcal F$ be a rank-one smooth $\overline{\mathbb F}_\ell$-sheaf on $U$. In a Cartesian diagram of smooth separated schemes

$$\begin{CD} V@>{\subset}>>Y\\ @VfVV @VV{\overline f}V\\ U@>{\subset}>>X \end{CD}$$

assume $U\subset X$ and $V\subset Y$ are complements of simple-normal-crossings divisors, $f:V\to U$ is a connected finite étale Galois covering trivializing $\mathcal F$, and $\mathcal F$ is clean with respect to $X$. Rank-one Swan class conjecture. (1) The equality

$$\operatorname{Sw}_{V/U,Y}(\mathcal F)=\overline f^*c_{\mathcal F,X}$$

in $CH_0(E\times_XY)\otimes_{\mathbb Z}\mathbb Q$ holds. (2) There exists a Cartesian diagram

$$\begin{CD} U@>{\subset}>>X'\\ @| @VV{\overline f}V\\ U@>{\subset}>>X \end{CD}$$

with $\overline f:X'\to X$ proper, $X'$ smooth over $F$, $U$ the complement of a simple-normal-crossings divisor in $X'$, and $\mathcal F$ clean with respect to $X'$. The first assertion identifies the abstract Swan class with Kato's refined rank-one cycle class, while the second is an expected alteration/existence statement; the supplied text gives no resolution.