Given a trivalent plane graph embedded in the Euclidean plane (up to isotopy), Treumann and Zaslow constructed and studied a certain associated Legendrian surface embedded in standard contact R5, nowadays referred to as a Legendrian 2-weave. Using gradient flow trees, Casals and Murphy computed its Legendrian contact dg-algebra with commutative coefficients (i.e. working over the group ring of the first homology group). We extend their computation to the non-commutative setting (i.e. working over the group ring of the fundamental group). For these Legendrian 2-weaves, we further verify the well-known conjecture that the moduli space of representations of this fully non-commutative version of the Legendrian contact dg-algebra are in bijective correspondence with a certain moduli space of sheaves.