Unique travelling-wave shooting-parameter conjecture for low ECM density

Fix $\kappa>0$ and define

$$m^*(\kappa)=\frac{1}{\kappa+1}.$$

Let $\bar{m}\in(0,m^*(\kappa)]$ and $c\geq2\sqrt{1-\bar{m}}$. For the travelling-wave system in the source, let $\alpha_0(c)$ and $\alpha_1(c)$ be the specified shooting-parameter bounds, and let $(n_{\alpha,c},p_{\alpha,c},m_{\alpha,c})$ denote the solution satisfying the asymptotic properties in Lemma 3.1. Unique shooting-parameter conjecture. There exists a unique $\alpha'\in[\alpha_0(c),\alpha_1(c))$ such that

$$(n_{\alpha',c},p_{\alpha',c},m_{\alpha',c})\longrightarrow(0,0,\bar{m})\qquad\text{as }y\to+\infty.$$

In particular, if $c=2\sqrt{1-\bar{m}}$, then $\alpha'=\alpha_0(c)$. The claim would establish uniqueness of the travelling wave in the stated low-density regime, conditional on the preceding concavity conjecture.