The difference $\tilde{\alpha}_{1}-\bar{\alpha}_{1}$ satisfies the following estimate: if either $y^{2}+a_{22}^{-1}\mu_{1}^{2}\gtrsim 1$ or $\mu_{2}\lesssim-A^{1/4}$, namely if $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}$, then $|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|\leq CA^{-1/4}$. Similar bounds hold for $\tilde{\alpha}_{i}-\bar{\alpha}_{i}$ for $i=1,2,3$.