Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

Let M1,…,Mn be right modules over a ring R. Suppose that the endomorphism ring of each module Mi has at most two maximal right ideals. Is it true that every direct summand of M1⊕⋯⊕Mn is a direct sum of modules whose endomorphism rings also have at most two maximal right ideals? We show that the answer is negative in general, but affirmative under further hypotheses. The endomorphism ring of uniserial modules, that is, the modules whose lattice of submodules is linearly ordered under inclusion, always has at most two maximal right ideals, and Pavel Příhoda showed in 2004 that the answer to our question is affirmative for direct sums of finitely many uniserial modules.