= Solution
For a stable one-species <linear noise approximation>, the <normalized stationary fluctuation-dissipation relation for a reaction network> becomes
$$
\boxed{\eta=\frac{\operatorname{Var}x}{m^2}
=\frac{\langle r\rangle}{Hm}.}
$$
To see the factors explicitly, the linearized restoring rate is $a=H/\tau=4\beta m$, while the chemical noise intensity is the rate-weighted squared step size
$$
B=\lambda\cdot1^2+\beta m^2\cdot(-2)^2=3\lambda.
$$
Stationarity of the <variance> gives $2a\operatorname{Var}x=B$. Using $\lambda=2\beta m^2$ therefore yields
$$
\operatorname{Var}x=\frac{3m}{4},\qquad
\boxed{\eta=\frac3{4m}=\frac3{4\langle x\rangle}.}
$$
This is the stationary fluctuation result within the stated negligible-fluctuation/linear-noise approximation, not an exact identity for the boundary-corrected discrete chemistry.
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