  For a simply-connected compact semisimple Lie group $G$ and its maximal torus
$T$, we study the $A_{\infty}$-functor associated to the moment Lagrangian
correspondence from the cotangent bundle $T^*G$ to the square $G/T^{-} \times
G/T$. In particular, we compute the leading term of the
$A_{\infty}$-homomorphism from the wrapped Floer cohomology $HW^*(T^*_e G,
T^*_e G)$ of the cotangent fiber $T_e^*G$ to the Floer cohomology $HF^*(\Delta,
\Delta)$ of the diagonal $\Delta$ in the square $G/T^{-} \times G/T$ by
determining the count of certain pseudo-holomorphic quilts. As a consequence,
we prove that the Floer cohomologies $HW^*(T^*_e G, T^*_e G)$ and
$HF^*(\Delta,\Delta)$ are isomorphic as rings after a localization.
