${\mathcal{I}}^{\mathrm{\prime }}$-curvatures in higher dimensions and the Hirachi conjecture

We construct higher-dimensional analogues of the ${\mathcal{I}}^{\mathrm{\prime }}$-curvature of Case and Gover in all CR dimensions $n\ge 2$. Our ${\mathcal{I}}^{\mathrm{\prime }}$-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 $n\ge 2$.