Specifically, if $\rho$ is such that $\zeta(\rho)=0$ and $\max_{\rho}\Re(\rho)= \Theta$, can anything interesting be said about the number/distribution of zeros on the vertical line $\sigma=\Theta$?

Clearly this question is almost as hypothetical as they get, so I welcome conditional answers (though not on RH please), consequences of the Bohr-Landau theorem, consequences of the known behavior of $\zeta(s)$ in the critical strip, etc.

Maybe you know something along the lines of ``If there are finitely/ infinitely many, then...''?

I am also interested in why your answer may simply be ``No.''