Vanishing conjecture for the hard-edge tacnode kernel

Let $\alpha>-1$, let $x,y\in\mathbb{R}_+$ be fixed, and let $\tau\in\mathbb{R}$ be fixed. The function $K^{\operatorname{tac},(\alpha)}(x,y;s,\tau)$ is the hard-edge tacnode kernel. Vanishing conjecture.

$$\lim_{s\to\infty}K^{\operatorname{tac},(\alpha)}(x,y;s,\tau)=0.$$

The conjecture asserts decay of the hard-edge tacnode kernel as the parameter $s$ tends to infinity, for every $\alpha>-1$ and fixed $x,y,\tau$. The supplied context gives no resolution, so its status remains open.