Legendrian Gronwall conjecture for Legendrian 3-webs

Let $M$ be a connected contact three-manifold. A Legendrian 3-web on $M$ is a set of three pairwise transversal foliations by Legendrian curves. A local linearization is a contactomorphism from an open subset of $M$ into $\mathbb{P}^3$ that maps every leaf of the foliations to a Legendrian line.

Legendrian Gronwall conjecture. Every Legendrian 3-web admits at most one distinct local linearization in $\mathbb{P}^3$, except when it is locally equivalent to the dual 3-web of the Legendrian twisted cubic curve in $\mathbb{P}^3$; in that exceptional case it admits exactly two distinct local linearizations.

This is the proposed Legendrian analogue of the classical Gronwall conjecture for planar 3-webs, which relates failure of unique linearization to algebraicity. The source presents the assertion as an open conjecture; it also states that the number of local linearizations is uniformly bounded and proves rigidity under suitable flatness assumptions.