Given two planar, conformal, smooth open sets
Ω
\Omega
and
ω
\omega
, we prove the existence of a sequence of smooth sets
(
Ω
ϵ
)
(\Omega _\epsilon )
which geometrically converges to
Ω
\Omega
and such that the (perimeter normalized) Steklov eigenvalues of
(
Ω
ϵ
)
(\Omega _\epsilon )
converge to the ones of
ω
\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.