It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, **do they also say something about the version of the Riemann Hypothesis in "Weil II"?** In particular, can the Standard Conjectures be "souped up" to get a version that does imply Weil II RH?

It seems that the main difference, in a nutshell, is that Weil II releases many of the assumptions on smoothness/properness/projectivity in the main results and considers cohomology with coefficients on a much more general class of sheaves. In particular, this implies that while the eigenvalues of Frobenius remain Weil numbers, they no longer necessarily have the same modulus for a given cohomology space. As a secondary question, are there any other major issues moving from Weil I to Weil II that I am missing here?