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In this paper, we prove interior a priori first- and second-order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that $u$ is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We also exhibit an explicit example showing that horizontal $W^{2,q}$ regularity of Calder\'on-Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general achieved in the range $q
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