= Solution
From the preceding conservation identity,
$$
\boxed{m_- =s^*-(s_-+v_-)=\frac{s^*-s_-}{1+\eta}=\frac1{1+\eta}\log\frac{s^*}{s_-}.}
$$
The nontrivial root $s_-$ depends only on $s^*$, through $\log(s^*/s_-)=s^*-s_-$. Therefore the coefficient $s^*-s_-$ is independent of $\eta$, proving the exact requested proportionality to $(1+\eta)^{-1}$, conditional on existence of the positive wave. This is stronger than an asymptotic estimate and differs from the spatially homogeneous problem with a fixed positive initially infected count.
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