Does assumption* imply strong orderability?

Determine whether, for a closed contact manifold (M,\xi), the condition assumption*—namely, the existence of a strong symplectic filling by a weakly^+-monotone symplectic manifold whose symplectic homology unit is not eternal—implies that (M,\xi) is strongly orderable. Establishing this implication would show that the contact big fiber theorem proved under assumption* also follows from the framework requiring strong orderability.

Background

Assumption* is used in prior work to derive contact big fiber theorems via symplectic homology, and it is known to imply orderability. The current paper’s approach yields a big fiber theorem under the stronger hypothesis of strong orderability, achieved through spectral selectors on strongly orderable manifolds.

Clarifying whether assumption* implies strong orderability would bridge the two approaches and potentially unify the respective big fiber results.