The procedure of many hypotheses logarithmically asymptotically optimal (LAO) testing for a model consisting of three or more independent objects is analyzed. It is supposed that M probability distributions are known and each object follows one of them independently of others. The matrix of asymptotic interdependencies (reliability-reliability functions) of all possible pairs of the error probability exponents (reliabilities) in optimal testing for this model is studied. This problem was introduced (and solved for the case of two objects and two given probability distributions) by Ahlswede and Haroutunian; it is a generalization of two hypotheses LAO testing problem for one object investigated by Hoeffding, Csiszár and Longo, Tusnády, Longo and Sgarro, Birgé, and others.