  We introduce an $A_\infty$ map from the cubical chain complex of the based
loop space of Lagrangian submanifolds with Legendrian boundary in a Liouville
Manifold $C_{*}(\Omega_{L} \mathcal{L}\mathit{ag})$ to wrapped Floer cohomology
of Lagrangian submanifold $\mathcal{CW}^{-*}(L,L)$. In the case of a cotangent
bundle and a Lagrangian co-fiber, the composition of our map with a previously
constructed map from $\mathcal{CW}^*(L,L) \to C_{*}(\Omega_q Q) $ shows that
this map is split surjective.
