Instability windows of relativistic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>r</mml:mi></mml:math>-modes in stably stratified neutron stars with hyperonic cores

(abridged) $R$-modes are oscillations in rotating stars, primarily restored by the Coriolis force. These oscillations are the most susceptible to the Chandrasekhar-Friedman-Schutz (CFS) instability driven by gravitational wave emission, which makes them promising targets for current and future gravitational wave searches. In order to develop, the instability must overcome dissipative processes within the star. As a result, $r$-modes become unstable only for certain combinations of stellar angular velocity $Ω$ and (redshifted) temperature $T^\infty$, defining the so-called instability window on the $(Ω, T^\infty)$ plane. At high temperatures, bulk viscosity $ζ$, arising from out-of-equilibrium chemical reactions, is the dominant dissipative agent. Dissipation due to $ζ$ can be greatly enhanced by two independent mechanisms: (1) the presence of hyperons, which significantly increases the bulk viscosity, and (2) the distinctive properties of relativistic $r$-modes in nonbarotropic matter, which further amplify dissipation beyond Newtonian predictions. In this work, we present the first investigation of the combined impact of these mechanisms on $r$-mode instability windows. Our calculations also account for the fact that chemical reactions modify the adiabatic index, in addition to producing bulk viscosity. We further estimate the influence of nucleon pairing effects on the instability windows. By comparing our predictions with recent observations of neutron stars in low-mass X-ray binaries, we find that bulk viscosity in hyperonic matter may provide the necessary dissipation to stabilize $r$-modes in the fastest-spinning and moderately hot stars, even when nucleon superfluidity and superconductivity are taken into account. These results have important implications for the interpretation of observations and for the broader understanding of relativistic $r$-mode physics.