= Solution
Let $m_i=\langle x_i\rangle$. The mean equations give $m_1=\lambda_1/\beta_1$ and $m_2=\lambda_2m_1/\beta_2$. For relative deviations, the drift Jacobian and normalized reaction-noise <covariance> are
$$
A=\begin{pmatrix}-\beta_1&0\\\beta_2&-\beta_2\end{pmatrix},\qquad D=\operatorname{diag}(2\beta_1/m_1,2\beta_2/m_2).
$$
The <normalized stationary fluctuation-dissipation relation for a reaction network> is $A\eta+\eta A^T+D=0$ with this negative-drift convention. Its entries successively give
$$
\boxed{\eta_{11}=\frac1{m_1},\qquad \eta_{12}=\frac{\beta_2}{\beta_1+\beta_2}\frac1{m_1},\qquad
\eta_{22}=\frac1{m_2}+\frac{\beta_2}{\beta_1+\beta_2}\frac1{m_1}.}
$$
The birth and death channels act on one species at a time, so the diffusion <matrix> is diagonal even though the stationary <covariance> is not.
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