The Burns–Epstein invariant formula conjecture for strictly pseudoconvex CR manifolds

Fix a positive integer $n$. Let $(M,T^{1,0}M)$ be a closed strictly pseudoconvex CR manifold of dimension $2n+1$ admitting a pseudo-Einstein contact form. For each partition $\varsigma\in\operatorname{Part}(n)$, let $\mathscr{I}_{\varsigma}(M)$ denote the corresponding global CR invariant, and let $\mu(M)$ denote the Burns–Epstein invariant. Burns–Epstein invariant formula conjecture. There exists a family $(C_{\varsigma})_{\varsigma\in\operatorname{Part}(n)}$ of real numbers, depending only on $\varsigma$, such that

$$\mu(M)=\sum_{\varsigma\in\operatorname{Part}(n)}C_{\varsigma}\mathscr{I}_{\varsigma}(M).$$

The paper establishes this type of relation for closed Sasakian $\eta$-Einstein manifolds and computes the coefficients in low dimensions. The conjecture proposes that the same universal linear relation holds for all closed strictly pseudoconvex CR manifolds admitting a pseudo-Einstein contact form.