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?



Do we know an upper bound for the number of possible real parts of the non trivial zeroes of $\zeta$?
This is for example mentioned as open in [a 2010 answer](https://mathoverflow.net/q/41120) by Fedor Petrov to basically a duplicate question.


It might be worth noting that nothing new has been proven in these last 6 years, or it would have been big news.


