Riemann's Auxiliary Function. Right limit of zeros

Numerical data suggest that the zeros $\rho$ of the auxiliary Riemann function in the upper half-plane satisfy $\mathop{\mathrm{Re}}(\rho)<1$. We show that this is true for those zeros with $\mathop{\mathrm{Im}}(\rho)> 3.9211\dots10^{65}$. We conjecture that this is true for all of them.