For a semigroup term \(f(x,y)\) in the variables \(x\) and \(y\), let \(\sigma_f\) stand for the corresponding hypersubstitution. The authors prove that the largest \(M\)-solid semigroup variety \(V_M\) (where \(M\) is a monoid of hypersubstitutions) is locally finite if and only if \(M\) contains one of the hypersubstitutions \(\sigma_{xyx}\) or \(\sigma_{yxy}\) (Theorem~3.10). The proof relies on an elegant application of \textit{M.~V.~Sapir}'s [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 2, 319--340 (1987; Zbl 0646.20047)] profound characterization of locally finite semigroup varieties of finite axiomatic rank. As a corollary of their main result, the authors deduce that the variety \(V_M\) is finitely based whenever \(M\) contains one of the hypersubstitutions \(\sigma_{xyx}\) or \(\sigma_{yxy}\) (Corollary~3.11). Reviewer's remark. In the abstract, the authors claim that the five-element Brandt semigroup \(B\) is not finitely based, referring to \textit{P.~Perkins} [J. Algebra 11, 298--314 (1969; Zbl 0186.03401)]. This is a mistake. P.~Perkins only proved that the six-element monoid obtained from \(B\) by adjoining to it an identity element is not finitely based while the semigroup \(B\) itself is known to be finitely based [see \textit{A.~N.~Trakhtman}, Issled. Sovrem. Algebre, Mat. Zap. 12, No. 3, 147--149 (1981; Zbl 0493.20038)].