Continuum of allosteric actions for non-amenable surface groups

Let $\Sigma$ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of p.m.p. ergodic minimal profinite actions for the fundamental group of $\Sigma$, that are topologically free but not essentially free, a property that we call allostery. Moreover, the IRS's we obtain are pairwise distincts.