The l-independence conjecture for traces of correspondences on smooth proper varieties

Let $K$ be a nonarchimedean local field with residue characteristic $p$, let $X/K$ be a proper smooth variety purely of dimension $n$, let $\sigma \in W_K$, and let $\Gamma \in CH^n(X \times_K X)$ be an algebraic correspondence. For each prime $l$, write $\operatorname{WD}(H^i(X \times_K K^{ac}, {\mathbb Q}_l^{ac}))$ for the associated Weil–Deligne representation. The l-independence conjecture. The alternating sum

$$\sum_{i=0}^{2n}(-1)^i\operatorname{tr}\bigl(\sigma\Gamma^*\big|\operatorname{WD}(H^i(X \times_K K^{ac},{\mathbb Q}_l^{ac}))\bigr)$$

lies in $\mathbb Q$ and is independent of $l$. Here $\sigma$ acts through the right action on $X\times_KK^{ac}$, and $\Gamma^*$ is the endomorphism defined by $\operatorname{pr}_{1,*}\circ([\Gamma]\cup)\circ\operatorname{pr}_2^*$. This is a standard expected compatibility between local and global cohomological realizations; the supplied text gives no resolution status.