The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem

We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking $S$-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szeg\"o class; on the other hand the transmission coefficient remains the same. This implies the statement given in the title.