Neumann monotonicity conjecture with volume scaling

Let $C(\Theta)$ be the geodesic ball of radius $\Theta$ in the constant-curvature model, let $\mu_k(\Theta)$ be its $k^{\text{th}}$ Neumann eigenvalue, and let $V(\Theta)$ denote its volume.

Neumann monotonicity with volume scaling. The function

$$\mu_k(\Theta)V(\Theta)^{2/n}$$

increases strictly for $\Theta\in(-\infty,\pi)$, for $k\geq2$ and $n\geq2$.

In dimension $n=2$, the paper reports increasing continuous behavior for the second Neumann eigenvalue, from $2\pi$ to $8\pi$. The conjecture proposes the corresponding higher-eigenvalue and higher-dimensional monotonicity.