By Lemma 2.7 the sum $\beta_{1}+\beta_{2}+\beta_{0}$ is independent of $\mu_{1},\mu_{2}$. Given $\eta\neq 0$, we shall evaluate this sum at the limit point $(\mu_{1}\to-\infty,\mu_{2}\to-\infty,\mu_{1}-\mu_{2}\to+\infty)$. Then $\beta_{0}$ has no contribution, while $\beta_{1}$ contributes

$$2\lim_{\mu_{1}-\mu_{2}\to+\infty}\int_{\mu_{1}=+\infty,\text{fix $\mu_{1}-\mu_{2}$}}^{\mu_{1}=-\infty}\frac{\partial\alpha_{1}}{\partial{\eta}}d\mu_{1},$$

and $\beta_{2}$ contributes

$$2\lim_{\mu_{1}\to-\infty}\int_{\mu_{2}=+\infty,\text{fix $\mu_{1}$}}^{\mu_{2}=-\infty}\frac{\partial\alpha_{2}}{\partial{\eta}}d\mu_{2}$$

plus

$$-2\lim_{\mu_{1}\to-\infty}\int_{\mu_{1}-\mu_{2}=-\infty,\text{fix $\mu_{1}$}}^{\mu_{1}-\mu_{2}=+\infty}\frac{\partial\alpha_{3}}{\partial{\eta}}d(\mu_{1}-\mu_{2}).$$

Observe

$$\lim_{\mu_{1}\to-\infty,\text{fix $\mu_{1}-\mu_{2}$}}\alpha_{3}(\mu_{1},\mu_{2},\eta)=\frac{1}{2\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2}}},$$

$$\lim_{\mu_{1}\to-\infty,\text{fix $\mu_{1}-\mu_{2}$}}\frac{\partial\alpha_{3}}{\partial\eta}=\frac{-(a_{11}+2a_{12}+a_{22})\bar{\eta}}{4((\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2})^{3/2}}$$

Using Lebesgue dominated convergence theorem, the third integral contribution is equal to

$$\frac{(a_{11}+2a_{12}+a_{22})\bar{\eta}}{2}\int_{-\infty}^{+\infty}\frac{1}{(x^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2})^{3/2}}dx=\frac{1}{\eta}.$$

The other two contributions are zero by similar arguments. ∎