  Dirac's conjecture, that secondary first-class constraints generate
transformations that do not change the physical system's state, has various
counterexamples. Since no matching gauge conditions can be imposed, the Dirac
bracket cannot be defined, and restricting the phase space first and then
quantizing is an inconsistent procedure. The latter observation has discouraged
the study of systems of this kind more profoundly, while Dirac's conjecture is
assumed generally valid. We point out, however, that secondary first-class
constraints are just initial conditions that do not imply Poisson's bracket
modification, and we carry out the quantization successfully by imposing these
constraints on the initial state of the wave function. We apply the method to
two Dirac's conjecture counterexamples, including Cawley's iconical system.
