Conjecture on oscillations of the real-line fractional KdV Green's function

Let $c>0$ and $\alpha\in(2,4]$, and let $G_{\mathbb{R}}$ denote the Green's function on the real line $\mathbb{R}$.

Real-line Green's function conjecture. For every $c>0$ and every $\alpha\in(2,4]$, $G_{\mathbb{R}}$ is not strictly positive on $\mathbb{R}$ and is not monotonically decreasing on $(0,\infty)$. It has a finite number of zeros on $\mathbb{R}$ if $\alpha\in(2,4)$ and an infinite number of zeros if $\alpha=4$.

This conjecture concerns the oscillatory behavior of the real-line Green's function that governs interactions of strongly localized waves. It is presented as a consequence of the preceding periodic-domain conjecture under the rescaling limit $c\to\infty$; no resolution is supplied in the source.