Stability and instability issues of the Weinstock inequality

Given two planar, conformal, smooth open sets $\mathrm{\Omega }$ and $\omega$, we prove the existence of a sequence of smooth sets ${\mathrm{\Omega }}_n$ which geometrically converges to $\mathrm{\Omega }$ and such that the (perimeter normalized) Steklov eigenvalues of ${\mathrm{\Omega }}_n$ converge to the ones of $\omega$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.