In fact, that is more than a conjecture, and it is always satisfied, regardless of whether or not there exist non-trivial zeros off the critical line. In this arxiv.org/pdf/0907.2426v5.pdf I have been studying a ratio strictly related to the inverse of your F(s). Said ratio is taken between the corresponding Dirichlet eta functions. It is graphically represented in Fig. 2, which also shows the upper and lower bound "sheets".