Isoperimetric inequality under Measure-Contraction property

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is an essentially non-branching metric measure space with $\mathfrak{m}(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N\in (1,\mathrm{\infty })$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality `a la L'evy-Gromov holds true. Measure theoretic rigidity is also obtained.