Finite convergence of the Moment-SOS hierarchy under hidden convexity

One considers polynomial optimization problems with compact feasible set $\mathbf{\Omega}$ defined by SOS-concave polynomials $g_j$, and with a globally non-convex polynomial objective $f$. We show that if $f$ is strongly convex on $\mathbf{\Omega}$, or SOS-convex on $\mathbf{\Omega}$ when the constraints $g_j$ are at most quadratic, then the associated Moment-SOS hierarchy converges in finitely many steps, without \`a priori knowledge of this hidden (local) convexity. In addition, in the latter case, the exact order for which the relaxation is exact is provided by the degree of a Putinar-like certificate of convexity. This demonstrates that a general-purpose hierarchy can adapt to favorable hidden properties of a specific instance without being informed of them, yielding certified global minimizers.