Conjecture on the limiting Evans function for regular internal solitary waves

Let $U^c(\xi,y)$ be a regular internal solitary wave (ISW), and let $D_N$ denote its $N$-th order truncated Evans function. Write $\mathbb{C}_+=\{\kappa\in\mathbb{C}:\operatorname{Re}\kappa>0\}$ and let $\overline{\mathbb{C}_+}$ be its closure. Limiting Evans-function conjecture. After suitable normalization of $D_N$, the limit

$$D=\lim_{N\to\infty}D_N$$

exists on $\overline{\mathbb{C}_+}$ and satisfies

$$D(0)=0,\qquad D'(0)=0.$$

If the amplitude of $U^c$ is sufficiently small, then additionally

$$D”(0)\ne 0,\qquad D(\kappa)\ne 0\quad\text{for all }\kappa\in\overline{\mathbb{C}_+}\setminus\{0\}.$$

This conjecture would establish a limiting Evans function with precisely the expected double zero at the origin and no other zeros in the closed right half-plane for sufficiently small-amplitude waves; its status is not resolved in the supplied text.