Bourgeois–Ekholm–Eliashberg cyclic surgery conjecture

Let $W$ be a Liouville cobordism corresponding to attaching $\ell\ge 1$ critical handles of index $k=n$ along a collection $\Lambda$ of disjoint Legendrian spheres. Let $L{\mathbb H}^{\text{cyc}}(\Lambda)_*$ denote the cyclic Legendrian contact homology group associated with $\Lambda$. With rational coefficients, there is an isomorphism

$$SH_*^{S^1,>0}(W,\partial^-W)\cong L{\mathbb H}^{\text{cyc}}(\Lambda)_*.$$

Bourgeois–Ekholm–Eliashberg cyclic surgery conjecture. Under this isomorphism, the surgery exact triangle for cyclic Legendrian contact homology is identified with the exact triangle of the pair $(V',V)$ for $SH_*^{S^1,>0}$. This is the $S^1$-equivariant, positive symplectic-homology formulation of the critical-handle surgery statement; the source presents it as conjectural because the underlying surgery exact triangle was not yet fully proved.