 Solution

Eliminate ${\mu }_2$ from the two connection formulas:At $(0,T)$, where ${\mu }_1=I$ and $\theta =-4k^{3}T$, this is the half-line mKdV spectral global relationTaking its upper-right entry gives a particularly useful scalar form,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 $e^{8i{k}^{3}T}$; 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.