The two-orientation conjecture for strongly quasipositive links

Let $L$ be a non-splittable unoriented link in $S^3$. An orientation of $L$ is strongly quasipositive if the oriented link bounds a strongly quasipositive surface. Two-orientation conjecture. $L$ has at most two strongly quasipositive orientations.

This conjecture arises from the preceding corollary concerning compatible contact structures of fibered Seifert links. The supplied text does not state any resolution or partial result beyond this motivation, so the conjecture remains open.