The nonnegative Laguerre-integral conjecture

Let $n\ge1$, $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$, and $a=(a_1,\dots,a_n)\in(0,+\infty)^n$ be as in the Laguerre-integral conjecture, and let $K_\alpha(a)$ denote the corresponding integral. Nonnegative Laguerre-integral conjecture. One has

$$K_\alpha(a)\ge0.$$

This is a weaker consequence of the sharper lower bound above and is equivalent in the paper to the operator inequality $\operatorname{Op}^w(\mathbf 1_{D_{a_1,\dots,a_n}})\le1$. It remains open in the anisotropic setting, although several special cases are proved.