= Solution
Eliminate $\mu_2$ from the two connection formulas:
$$
\mu_3=\mu_1e^{i\theta\widehat\sigma_3}(S^{-1}s).
$$
At $(0,T)$, where $\mu_1=I$ and $\theta=-4k^3T$, this is the <half-line mKdV spectral global relation>
$$
\boxed{S^{-1}s=e^{4ik^3T\widehat\sigma_3}\mu_3(0,T,k).}
$$
Taking its upper-right entry gives a particularly useful scalar form,
$$
\boxed{A(k)b(k)-B(k)a(k)=e^{8ik^3T}c(k,T),\qquad c=\mu_{3,12}(0,T,k).}
$$
The second column of $\mu_3$ shows that $c$ is analytic in the lower half-plane and has the usual $O(k^{-1})$ normalization. The opposite off-diagonal equation follows from the real-$q$ reduction. The exponential sign here follows from conjugating a $12$ entry, which multiplies it by $e^{8ik^3T}$; reversing it would give the wrong linear <global relation> and boundary count. This relation constrains the initial and boundary <scattering data>; they cannot be chosen independently.
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