Swan trace formula for tensor products over finite fields

Let $U$ be a connected separated smooth scheme of dimension $d$ of finite type over a finite field $F$, let $X$ be a proper normal scheme over $F$ containing $U$ as a dense open subscheme, and let $Fr_F\in\operatorname{Gal}(\overline F/F)$ be geometric Frobenius. Let $\rho_X:CH_0(X)\to\pi_1(X)^{\mathrm{ab}}$ be the reciprocity map sending a closed point class $[x]$ to geometric Frobenius $Fr_x$. Let $\ell$ be invertible in $F$, let $\mathcal F$ be a smooth $\overline{\mathbb F}_\ell$- or $\overline{\mathbb Q}_\ell$-sheaf on $U$, and assume the integrality conjecture holds and $\operatorname{Sw}_X(\mathcal F)\in CH_0(X\setminus U)$ is defined. For a smooth sheaf $\mathcal G$ on $X$, let $\det\mathcal G:\pi_1(X)^{\mathrm{ab}}\to\overline{\mathbb F}_\ell^\times$ or $\overline{\mathbb Q}_\ell^\times$ be the character corresponding to its determinant. Define

$$\det(-Fr_F:H_c^*(U_{\overline F},\mathcal F))=\prod_{q=0}^{2d}\det(-Fr_F:H_c^q(U_{\overline F},\mathcal F))^{(-1)^q}.$$

Swan trace formula conjecture. Then

$$\det(-Fr_F:H_c^*(U_{\overline F},\mathcal F\otimes\mathcal G)) =\det(-Fr_F:H_c^*(U_{\overline F},\mathcal F))^{\operatorname{rank}\mathcal G}\cdot\det\mathcal G\bigl(\rho_X(\operatorname{Sw}_X(\mathcal F))\bigr).$$

This is presented as an expected refinement of the theorem cited in the source; the supplied text gives no resolution.