Write $\rho=\beta + i\gamma$ and $\rho'=\beta'+i\gamma'$ for two distinct non-trivial zeros of the Riemann zeta-function. Is it known that $\gamma\neq\gamma'$? That is, does a proof exist that two zeros can't have the same ordinate?
Certainly, it is true under the Riemann hypothesis but I'm curious as for whether this is true unconditionally.
EDIT: To clarify, by distinct I mean that $\rho\neq\rho'$ so that we are not considering any matters relating to multiplicity. Also, as mentioned in the comments one should assume $\beta,\:\beta'\geq\frac{1}{2}$ as to avoid triviality.