= Solution
Put $\epsilon=m_2/m_1\ll1$ and $t_{s2}/t_{s1}=\epsilon^k$, with $0<k<1$. The <gas-driven contraction of a planetesimal binary> and <gas-driven migration of a binary centre of mass> times obey
$$
\frac\tau{\tau_b}=\eta\frac{m_1/t_{s1}+m_2/t_{s2}}{m_1/t_{s2}+m_2/t_{s1}}
=\eta\frac{1+\epsilon^{1-k}}{\epsilon^{-k}+\epsilon}.
$$
Therefore
$$
\boxed{\frac\tau{\tau_b}\simeq\eta\epsilon^k\ll1.}
$$
The low-mass secondary has the shorter <aerodynamic stopping time> and chiefly controls internal damping, whereas the primary chiefly controls <centre of mass> drag. The pair can contract substantially, potentially merging, before its stellar orbital radius changes appreciably. This conclusion is restricted to a pair that remains bound and within the adopted linear-drag regime.
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