(Thomas-Yau uniqueness in the weak setting) Suppose $L,L^{\prime}$ are two special Lagrangian integral currents $L,L^{\prime}$ with the same phase angle $\hat{\theta}$, equipped with suitable unobstructed brane structures, such that $L\simeq L^{\prime}$ in $D^{b}Fuk(X)$. Then $L=L^{\prime}$ as currents.