  Given two planar, conformal, smooth open sets $\Omega$ and $\omega$, we prove
the existence of a sequence of smooth sets $\Omega_n$ which geometrically
converges to $\Omega$ and such that the (perimeter normalized) Steklov
eigenvalues of $\Omega_n$ 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.
