The eternal quotient dichotomy for symplectic cohomology

Let $W$ be a convex-at-infinity symplectic manifold. For each contact isotopy $\varphi_t$, let $\mathrm{HF}(\varphi_t)$ denote its Floer cohomology, and let $\mathrm{SH}^{e}(W)$ be the subspace of $\mathrm{SH}(W)$ generated by elements lying in the image of $\mathrm{HF}(\varphi_t)\to\mathrm{SH}(W)$ for every contact isotopy $\varphi_t$. The quotient $\mathrm{SH}(W)/\mathrm{SH}^{e}(W)$ is either zero or infinite dimensional. This dichotomy extends the aspherical case, where the quotient vanishes if and only if $\mathrm{SH}(W)$ vanishes; the conjecture concerns the possible presence of holomorphic spheres and the resulting quantum corrections.