Thank you @GHfromMO. My impression is that by assuming the karatsuba bounds on $|\zeta(s)|$ in infinitesimally short imaginary intervals, both EH and The classical Chebyshev conjecture follow.

much appreciated @Alessandro Languasco. The Selberg paper I know quite well, but had no idea about Gallagher and Mueller, or about Heath-Brown and Goldston. Glancing through "The notes on the pair...", you get a nice historical sense on the natural evolution of the ideas, something often missing in dry technical papers

@Sam Hopkins yes. Erdős also says, if I recall correctly, that the bound, if correct, would lie "quite deep", meaning that he saw it as important. This was actually what sparked my interest in the problem, to be honest, much more so than Selberg's paper

Thank you. I will also stop here. Your point is cristal clear, but my impression is that @bojonsson might be assuming $s$ is on the critical line, or strip. I have posted an answer with some basic comments.

Thank you @GHfromMO. Of course the bound is trivial for bounded p. But the differing speeds at which $\xi(s)$ and $f(s)$ grow to infinity faster than exponentially in the positive axis, is far from trivial...

But it seems to me that the problem is still open for circles of sufficiently small radius, since the behaviors of $\xi(s)$ in short intervals is still a mystery. The question is trivial for large p, I agree...

@GHfromMO Ok thank you. I see your point then. Still I'm not convinced that the bound is not satisfied, but I must be missing something...

It is well known that there exist certain entire functions of order 1, that satisfy the bound mentioned. That all entire functions of order 0 satisfy the bound, is also well known, as you mentioned. Are you sure that the xi function does not satisfy this bound? is this not an open problem still?