Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself. The proof uses a variety of techniques in contact and symplectic geometry, from open book decompositions and Weinstein handle diagrams, to symplectic homology and the Legendrian contact homology DGA.