The assumption that $-\pi/2+\epsilon<\inf_{L}\theta\leq\sup_{L}\theta<\pi/2-\epsilon$ for the minimizer is not innocent, but represents a principal gap in our program to find special Lagrangian currents. The problem is that if on the minimizer $\sup_{L}\theta_{L}=\frac{\pi}{2}-\epsilon$, and a priori $L$ has no regularity assumption (eg. the Lagrangian angle may a priori be highly oscillatory), then we lack techniques to construct Lagrangian competitors which remain quantitatively almost calibrated.