Monotonicity conjecture for the first Steklov eigenvalue of an eccentric annulus

Let $B_r(a,b)$ denote the ball of radius $r>0$ centered at $(a,b)$, and define

$$\Omega_{\epsilon}:=B_{1}(0,0)\setminus B_{\delta}(0,\epsilon).$$

Let $\sigma_1(\Omega_{\epsilon})$ be the first non-zero Steklov eigenvalue, with $\epsilon\in[0,1-\delta)$.

Monotonicity conjecture. The first non-zero Steklov eigenvalue $\sigma_1(\Omega_{\epsilon})$ is a monotone decreasing function of $\epsilon$ on $[0,1-\delta)$.

The conjecture is motivated by numerical computations for annuli whose inner disk moves inside the unit disk. The supplied text reports numerical validation of the discretization and presents this monotonicity claim, but gives no resolution beyond the numerical evidence.