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This paper considers adaptive detection and estimation in the presence of useful signal and interference mismatches. We assume a homogeneous environment where the random disturbance components from the primary and secondary data share the same covariance matrix. Moreover, the data under test contains a deterministic interference vector in addition to the possible useful signal. We focus on the situation where an energy fraction of both the useful signal and the deterministic interference may lie outside their nominal subspaces (conical uncertainty model). Under these conditions, we devise a procedure for the computation of the joint maximum likelihood (ML) estimators of the useful signal and interference vectors, resorting to a suitable rank-one decomposition of a semidefinite program (SDP) problem optimal solution. Hence, we use the aforementioned estimators for the synthesis of adaptive receivers based on different generalized likelihood ratio test (GLRT) criteria. At the analysis stage, we assess the performance of the new detectors in comparison with some decision rules, available in open literature.