There is a weighted partition $L=A_{1}+A_{2}$ such that

$$\text{Re}\int_{A_{i}}\Omega=\text{Re}\int_{L_{i}}\Omega>0,\quad\text{Im}\int_{A_{2}}\Omega\leq\text{Im}\int_{L_{2}}\Omega,\quad\text{Im}\int_{A_{1}}\Omega\geq\text{Im}\int_{L_{1}}\Omega.$$

Consequently $\arg\int_{A_{2}}\Omega\leq\arg\int_{L_{2}}\Omega=\hat{\theta}_{2}$ and $\arg\int_{A_{1}}\Omega\geq\arg\int_{L_{1}}\Omega=\hat{\theta}_{1}$, so in particular $\inf_{L}\theta_{L}\leq\hat{\theta}_{2}$ and $\sup_{L}\theta_{L}\geq\hat{\theta}_{1}$.