The author proposes an iterative procedure for the inverse Toeplitz eigenproblems: diagonalize the current approximation \(T_ k\) to the required matrix to determine the corresponding matrix of eigenvectors \(Q_ k\), then construct \(T_{k+1}\) as the Toeplitz matrix which has the given eigenvalues as its Rayleigh quotients associated with the eigenvectors \(Q_ k\). Although convergence could not be established, the author gives a counterexample to the conjecture of \textit{P. Delsarte} and \textit{Y. Genin} [Lect. Notes Control Inf. Sci. 58, 194-213 (1984; Zbl 0559.15017)] that the eigenvector of a symmetric Toeplitz matrix, corresponding to eigenvalues arranged in decreasing order, alternate between reciprocal and antireciprocal vectors.