Construction of a unique metric in quasi-Hermitian quantum mechanics: Nonexistence of the charge operator in a 2×2 matrix model

For a specific exactly solvable 2×2 matrix model with a PT-symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics Θ>0 and show that none of them admits a factorization Θ=CP in terms of an involutive charge operator C. Alternative ways of restricting the physical metric to a unique form are briefly discussed.