Simple state preparation for contextuality tests with few observables

We consider any noncontextuality inequality, and the state preparation scheme which consists in performing any von Neumann measurement, with post-measurement states rho(m), on any initial state. For an inequality which is not always satisfied and Hilbert space dimensions greater than a value specified by the inequality, we determine necessary and sufficient conditions for the existence of observables with which the inequality is violated for a state rho(m). For an initial state with no zero eigenvalues, there are always such observables, and they are independent of this state.