Dirichlet monotonicity conjecture with volume scaling

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

Dirichlet monotonicity with volume scaling. The function

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

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

The conjecture extends the known first-eigenvalue behavior and the established negative-radius behavior of the second eigenvalue, while its behavior on $(0,\pi)$ is not deduced by the cited results.