Strong-coupling scattering-matrix convergence conjecture for leaky curves

Let $\Gamma$ be a $C^4$-smooth, asymptotically straight planar curve. Let $\Sigma_{\alpha,\Gamma}(k)$ be the on-shell scattering matrix at energy $k^2$, and let $S_\Gamma$ be the one-dimensional comparison operator associated with $\Gamma$, with on-shell scattering matrix $\Sigma_{S_\Gamma}(k)$. Strong-coupling scattering-matrix convergence conjecture. For the strong-coupling limit $\alpha\to\infty$,

$$\Sigma_{\alpha,\Gamma}\left(k-\frac14\alpha^2\right)\longrightarrow\Sigma_{S_\Gamma}(k).$$

This expresses the expected effective one-dimensional behavior of scattering by a sufficiently smooth leaky curve in the strong-coupling regime; the source presents it as an expectation rather than a proved result.