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?