Conjectured eigenvalue monotonicity for an ellipse with an off-center hole

Let $\Omega_{out}$ be the ellipse $E_{r,R}$ in $\mathbb{R}^{2}$ centered at the origin, with minor axis $r$ and major axis $R$, 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 contained in the ellipse. The distance between the centers is the distance from the origin to $(t_1,t_2)$.

Ellipse-hole eigenvalue monotonicity conjecture. The eigenvalues $\sigma_1(\Omega_{(t_1,t_2)})$ and $\mu_2(\Omega_{(t_1,t_2)})$ decrease with respect to the distance between the centers of $E_{r,R}$ and $B_1(t_1,t_2)$. The eigenvalue $\mu_1(\Omega_{(t_1,t_2)})$ decreases with respect to that distance when the center of $B_1(t_1,t_2)$ varies along the lines $y=0$ and $y=x$, but increases when the center varies along the line $x=0$.

These claims are based on numerical observations for an ellipse as outer domain and are stated for the specified directions of motion. Their general validity is not established in the supplied text.