Maurer–Cartan deformation conjecture for ambient symplectic cochains

Let $D$ be a Liouville domain whose boundary $\partial D$ has no contractible Reeb orbits. The intrinsic symplectic cochains $SC^*(D)$ and the $S^1$-equivariant symplectic cochains $SC^*_{S^1}(D)$ are related by a Gysin map

$$\rho:SC^*_{S^1}(D)\to SC^{*+1}(D),$$

and $\rho_*$ denotes the induced map on Maurer–Cartan moduli spaces. Maurer–Cartan deformation conjecture. There is a Maurer–Cartan element $x\in SC^*_{S^1}(D)$, well-defined up to gauge equivalence, such that

$$SC^*_M(D)\simeq \text{the twist of }SC^*(D)\text{ by }\rho_*(x).$$

The conjecture is motivated by an SFT-type neck-stretching argument, which is expected to produce an element in linearized contact chains and then map it to positive equivariant symplectic cochains and ultimately to $SC^*_{S^1}(D)$.