  We construct higher-dimensional analogues of the
$\mathcal{I}^\prime$-curvature of Case and Gover in all CR dimensions $n\geq2$.
Our $\mathcal{I}^\prime$-curvatures all transform by a first-order linear
differential operator under a change of contact form and their total integrals
are independent of the choice of pseudo-Einstein contact form on a closed CR
manifold. We exhibit examples where these total integrals depend on the choice
of general contact form, and thereby produce counterexamples to the Hirachi
conjecture in all CR dimensions $n\geq2$.
