  There is an abundance of evidence that some relaxation dynamics, e.g.,
exponential decays, are much more common in nature than others. Recently, there
have been attempts to trace this dominance back to a certain stability of the
prevalent dynamics versus generic Hamiltonian perturbations. In the paper at
hand, we tackle this stability issue from yet another angle, namely in the
framework of the recursion method. We investigate the behavior of various
relaxation dynamics with respect to alterations of the so-called Lanczos
coefficients. All considered scenarios are set up in order to comply with the
"universal operator growth hypothesis". Our numerical experiments suggest the
existence of stability in a larger class of relaxation dynamics consisting of
exponentially damped oscillations. Further, we propose a criterion to identify
"pathological" perturbations that lead to uncommon dynamics.
