The Weil conjecture for the counting Euler characteristic

Let $(K_\infty,\sigma)$ be a model of $\operatorname{ACPF}$, let $V$ be a smooth projective variety over $K_1$, and let $\ell$ be a prime different from the characteristic. Define the $\ell$-adic cohomology by

$$H^i(V;\mathbb{Q}_\ell)=\mathbb{Q}_\ell\otimes_{\mathbb{Z}_\ell}\varprojlim_k H^i_{\mathrm{et}}(V\times_{K_1}K_\infty;\mathbb{Z}/\ell^k).$$

Weil conjecture. The $\ell$-adic component of $\chi(V(K_1))$ is

$$\sum_{i=0}^{2\dim(V)}(-1)^i\operatorname{Tr}\bigl(\sigma^{-1}\mid H^i(V;\mathbb{Q}_\ell)\bigr).$$

This would relate the $\mathbb{Z}_\ell$-valued counting Euler characteristic to $\ell$-adic étale cohomology. The paper lists this as a direction for future research and does not establish it.