What is a wave function? It is the solution of a quantum mechanical equation ( with the appropriate potentials),on which boundary conditions are imposed to make it specific to a system . $|\psi\rangle$ by itself is not independent of the environment the way that the operators X are.
Thus the answer depends on the system under consideration.
I like using the single electron at a time double slit experiment because it shows how the boundary conditions that will define $|\psi\rangle$ have to be taken into account.
The wavefunction we need is the solution of the topology :plane wave single electron , field of two slits. The operator in this case is the (x,y) operator that acted on the screen to give the dots on the top image.
For each individual electron the $|\psi\rangle$ that describes its probability changes the minute the operator X operates (hit on the screen) . A completely different $|\psi\rangle$ will describe it from then on because the fields and boundary conditions are drastically different. If there were no screen and the electron could propagate to infinity the wavefunction describing its probabilistic manifestation mathematically is not changed.
The addition of many electrons describe the probability distribution in (x,y) for this setup, i.e. the complex conjugate square of the wavefunction, up to the z of the screen, where everything changes.