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### Description

Detectability of the $r$-mode gravitational-wave signal depends on the interplay between the mode amplification by the CFS instability and its damping by dissipative mechanisms, operating in the stellar matter. Those stellar parameters - usually, the angular velocity, $\Omega$, and redshifted temperature, $T^\infty$, - for which the mode is unstable, define the $r$-mode instability window. We revisit this problem in nonbarotropic neutron stars, accounting for the previously overlooked relativistic $r$-mode nonanalytic behavior (in $\Omega$) and enhanced energy dissipation due to diffusion in superconducting stellar matter. We show that, at slow rotation rates, relativistic $r$-modes are amplified by the CFS instability weaker than Newtonian ones, while their viscous and diffusive dissipation is, instead, significantly more efficient. At realistic rotation rates relativistic and Newtonian $r$-mode amplification by CFS mechanism and damping by shear viscosity become comparable, while the relativistic mode damping by diffusion and bulk viscosity remain significantly stronger than nonrelativistic ones. As a result, accounting simultaneously for diffusion and relativistic $r$-mode nonanalyticity drastically changes the $r$-mode instability window as compared to the Newtonian one. This effect is of paramount importance for the interpretation of the future gravitational-wave observations and understanding of the $r$-mode physics in general.