We study the behaviour of sequences U2nXU1-n, where U₁, U₂ are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator XU₁- U₂X is small in a sense. The conjecture about the weak averaged convergence of the difference U2nXU1-n-U2-nXU1n to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where U₁= U₂ is the unitary operator of multiplication by z on L²(μ) , sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.