  The Gronwall conjecture states that a planar 3-web of foliations which admits
more than one distinct linearizations is locally equivalent to an algebraic
web. We propose an analogue of the Gronwall conjecture for the 3-web of
foliations by Legendrian curves in a contact three manifold. The Legendrian
Gronwall conjecture states that a Legendrian 3-web admits at most one distinct
local linearization, with the only exception when it is locally equivalent to
the dual linear Legendrian 3-web of the Legendrian twisted cubic in $\,\PP^3$.
We give a partial answer to the conjecture in the affirmative for the class of
Legendrian 3-webs of maximum rank. We also show that a linear Legendrian 3-web
which is sufficiently flat at a reference point is rigid under local linear
Legendrian deformation.
