Winding tachyons in asymptotically supersymmetric black strings

We show that in the D1-D5 system with angular momentum, there can be localised tachyonic winding string modes in the interior of the spacetime even if we choose a spin structure which preserves supersymmetry in the asymptotic region. We consider cases where the tachyonic region extends outside the event horizon, and argue that the natural endstate of tachyon condensation in almost all cases is one of the solitonic solutions which correspond to special microstates of the D1-D5 system.