Conjecture on the periodic fractional KdV Green's function

Let $\alpha\in(2,4]$ and $c>0$. The periodic Green's function $G_{\mathbb{T}}$ is defined by the formulas referred to in the source as (green) and (green-fourier).

Periodic Green's function conjecture. For each $\alpha\in(2,4]$, there exists $c_0>0$ such that, for $c\in(0,c_0)$, $G_{\mathbb{T}}$ is even, strictly positive on $\mathbb{T}$, and monotonically decreasing on $(0,\pi)$. For $c\in[c_0,\infty)$, $G_{\mathbb{T}}$ has a finite number of zeros on $\mathbb{T}$. The number of zeros remains bounded as $c\to\infty$ if $\alpha\in(2,4)$ and is unbounded as $c\to\infty$ if $\alpha=4$.

The conjecture extends the established comparison between the periodic Green's function for $\alpha\in(0,2)$ and the case $\alpha=2$ to the range $\alpha\in(2,4]$, where numerical results suggest a transition from positivity and monotonicity to oscillatory behavior as the wave speed parameter grows.