In this paper we deal with optimality conditions that can be verified by a nonlinear optimization algorithm, where only a single Lagrange multiplier is avaliable. In particular, we deal with a conjecture formulated in [R. Andreani, J.M. Martinez, M.L. Schuverdt, ``On second-order optimality conditions for nonlinear programming'', Optimization, 56:529--542, 2007], which states that whenever a local minimizer of a nonlinear optimization problem fulfills the Mangasarian-Fromovitz Constraint Qualification and the rank of the set of gradients of active constraints increases at most by one in a neighborhood of the minimizer, a second-order optimality condition that depends on one single Lagrange multiplier is satisfied. This conjecture generalizes previous results under a constant rank assumption or under a rank deficiency of at most one. In this paper we prove the conjecture under the additional assumption that the Jacobian matrix has a smooth singular value decomposition, which is weaker than previously considered assumptions. We also review previous literature related to the conjecture.