Nonnegativity conjecture for the truncated Bessel kernel

Let $K_M^\alpha(x)$ be the kernel defined in the paper, with $M\in\mathbb{N}$, $x\in[0,1]$, and real parameter $\alpha$. Numerical simulations suggest its nonnegativity for $0\leqslant\alpha\leqslant\frac{1}{2}$.

Nonnegativity conjecture. For all $0\leqslant\alpha\leqslant\frac{1}{2}$, $M\in\mathbb{N}$, and $x\in[0,1]$,

$$K_M^\alpha(x)\geqslant 0.$$

The claim is motivated by numerical simulations and is presented as a conjecture; no proof or disproof is supplied in the given text.