Smoothness and positivity conjecture for band-limited nodal-area distributions

Let $f=f_{\alpha;T}$ be a band-limited Gaussian function on an $n$-dimensional compact manifold, with $n\geq 2$ and $\alpha\leq 1$, and let $\Psi_{n;\alpha}$ be the limiting distribution function for normalized nodal-domain volumes. For $\alpha=1$, let $t_0=\frac{\pi^{n/2}}{\Gamma(n/2+1)}j^n_{n/2-1,1}$, where $j_{n/2-1,1}$ is the first zero of the Bessel function $J_{n/2-1}$. Smoothness and positivity conjecture. The function $\Psi_{n;\alpha}$ is continuous and everywhere differentiable. For $\alpha<1$, its derivative satisfies $\frac{d\Psi_{n;\alpha}(t)}{dt}>0$ everywhere, whereas for $\alpha=1$ the same inequality holds for $t>t_0$. This conjecture predicts that the limiting nodal-volume distribution has no atoms and a strictly positive density throughout its support, with the stated threshold in the monochromatic case.