Conjectured eigenvalue monotonicity for a ball with an off-center hole

Let $\Omega_{out}$ be the ball $B_{R_2}$ in $\mathbb{R}^{2}$ of radius $R_2$ centered at the origin, and let $\Omega_{(t_1,t_2)}=\Omega_{out}\setminus B_1(t_1,t_2)$, where $B_1(t_1,t_2)$ is a unit ball inside $\Omega_{out}$. The distance between the centers is the distance from $(0,0)$ to $(t_1,t_2)$.

Ball-hole eigenvalue monotonicity conjecture. The first and second nonzero Steklov eigenvalues $\sigma_1(\Omega_{(t_1,t_2)})$ and $\sigma_2(\Omega_{(t_1,t_2)})$ decrease as the distance between the centers increases. Among all positions of $B_1(t_1,t_2)$ inside $B_{R_2}$, both eigenvalues attain their maximum when the balls are concentric and their minimum when the inner ball touches the outer ball. The first nonzero mixed Steklov Neumann eigenvalue $\mu_1(\Omega_{(t_1,t_2)})$, with Steklov condition on the outer boundary and Neumann condition on the inner boundary, has multiplicity $2$ and decreases as the distance between the centers increases.

These claims are based on numerical observations for doubly connected domains with a ball as outer boundary. Their general validity is not established in the supplied text.