Hirachi's decomposition conjecture for pseudohermitian scalar invariants

Let $I$ be a natural pseudohermitian scalar invariant whose total integral is a secondary CR invariant. Here, a secondary CR invariant is a total integral invariant under changes of pseudohermitian contact form. Hirachi's conjecture. There is a constant $c\in\mathbb{R}$ such that

$$I=cQ^\prime+\text{(local CR invariant)}+\text{(divergence)}.$$

The conjecture is an analogue in CR geometry of the decomposition result for conformal scalar invariants, with the total $Q^\prime$-curvature playing the role of total $Q$-curvature. It is false in CR dimension $n=2$, so the proposed decomposition does not hold in the stated generality.