= Solution
Suppose $U_+=U$ and $U_-=-U$. Integrating the two velocity profiles gives the <volume fluxes> per unit span
$$
Q_+=d_+U+\frac{\tau d_+^2}{2\mu_+},
\qquad
Q_-=-d_-U-\frac{\tau d_-^2}{2\mu_-}.
$$
Here
$$
\tau=\frac{-2U}{d_+/\mu_++d_-/\mu_-}.
$$
Consequently
$$
Q_+=\frac{Ud_+d_-/\mu_-}{d_+/\mu_++d_-/\mu_-},
\qquad
Q_-=-\frac{Ud_+d_-/\mu_+}{d_+/\mu_++d_-/\mu_-},
$$
so all dependence on the interface position cancels from the ratio:
$$
\boxed{\frac{Q_-}{Q_+}=-\frac{\mu_-}{\mu_+}.}
$$
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