  We calculate asymptotic estimates for the conjugacy growth function of
finitely generated class 2 nilpotent groups whose derived subgroup is infinite
cyclic, including the so-called higher Heisenberg groups. We prove that these
asymptotics are stable when passing to commensurable groups, by understanding
their twisted conjugacy growth. We also use these estimates to prove that, in
certain cases, the conjugacy growth series cannot be a holonomic function.
