  Meeks, P\'erez and Ros conjectured that a closed Riemannian $3$-manifold
which does not admit any closed embedded minimal surface whose two-sided
covering is stable, must be diffeomorphic to a quotient of the $3$-sphere. We
give an counterexample to this conjecture. Also, we show that if we consider
immersed surfaces instead of embedded ones, then the corresponding statement is
true.
