Higher-dimensional Neumann monotonicity conjecture

Let $C(\Theta)$ be the geodesic ball of radius $\Theta$ in the constant-curvature model, and let $\mu_k(\Theta)$ denote its $k^{\text{th}}$ Neumann eigenvalue. At $\Theta=0$, let $\mu_k(\mathbb{B}^n)$ denote the corresponding eigenvalue of the Euclidean ball $\mathbb{B}^n\subset\mathbb{R}^n$.

Higher-dimensional Neumann monotonicity conjecture. The function

$$\Theta \mapsto \begin{cases} \mu_k(\Theta)\,\sinh^2(\Theta), & \Theta\in(-\infty,0), \\ \mu_k(\mathbb{B}^n), & \Theta=0, \\ \mu_k(\Theta)\,\sin^2(\Theta), & \Theta\in(0,\pi/2), \end{cases}$$

decreases strictly from $\infty$ to $0$, for all $k\geq2$ and $n\geq3$.

The claim is supported by numerical evidence, but the proof used for the Dirichlet spectrum does not transfer readily to the Neumann case. The corresponding statement in dimension $n=2$ was proved by Langford and Laugesen.