The cohomological ell-independence conjectures

Let $X/F$ be a variety, let $i$ be a cohomological degree, and let H^i_\u0**eell(X) and $H^i_{c,\u0**eell}(X)$ denote the corresponding Weil--Deligne representations, with the latter having compact support. Cohomological ell-independence conjectures. The systems \{H^i_\u0**eell(X)\}_\u0**eell, \{H^i_\u0**eell(X)\}_\u0**eell weakly, \{H^i_{c,\u0**eell}(X)\}_\u0**eell, and \{H^i_{c,\u0**eell}(X)\}_\u0**eell weakly are respectively ${\mathbb Q}$-compatible, weakly ${\mathbb Q}$-compatible, ${\mathbb Q}$-compatible, and weakly ${\mathbb Q}$-compatible. These compatibility assertions formalize ell-independence for cohomology; the full compatibility statement remains wide open in general and would follow from Frobenius semisimplicity.