Let $n_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n_{\zeta}=1$, and the symmetry arising from the functional equation implies that $n_{\zeta}$ is odd.
My question is thus: do we know an upper bound for $n_{\zeta}$? Are we at least sure it is finite?