  Stationary measures on the circle that arise from a large class of random
walks on the fundamental group of a finite-area complete hyperbolic surface
with cusps are singular with respect to the Lebesgue measure. In particular, it
is sufficient for singularity that a stationary measure satisfies an
exponential decay for cusp excursions with excursions being measured in the
path metric on horocycles bounding cusps. In this note, we settle a conjecture
of McMullen by proving that the singularity of stationary measures satisfying
such exponential decay is quasi-symmetrically stable, that is under
push-forward by any quasi-symmetry of the circle the measure remains singular.
