Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures

We investigate the validity of the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) for sub-Riemannian structures beyond step two. We show that whenever the distance function is not Lipschitz in charts, the MCP may fail. This occurs already in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. Since the MCP is preserved under blow-ups, we focus on Carnot homogeneous spaces, proving that MCP descends to suitable quotients. As a byproduct, any structure whose tangent at some point admits a quotient to Martinet fails the MCP. We also obtain a computation-free proof that the Grushin plane shares the Heisenberg group's MCP. Applications include a detailed analysis of validity and failure of the MCP for Carnot groups of low dimension. Our results suggest a conjecture on the failure of the MCP in presence of Goh abnormal geodesics satisfying the strong generalized Legendre condition.