The extended canonical Noether identities derived from an extended action in the phase space for a system with a higher-order singular Lagrangian are formulated.Based on the canonical symmetries of generalized constrained Hamiltonian systems, a counterexample to a conjecture of Dirac is given. Using the canonical first Noether theorem and canonical Noether identities and the extended canonical Noether identities, we have shown that Dirac's conjecture fails for a system with a higher-order singular Lagrangian in which there is no linearization of constraint in our treatment.