  We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special
smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth
irreducible projective threefold if $X_{10}$ is general among special
Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We
construct an irreducible component of a moduli space of Bridgeland stable
objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of
$\mathcal{H}$. We also identify the minimal model of Fano surface
$\mathcal{C}(X_{10}')$ of conics on a smooth ordinary Gushel-Mukai threefold
with moduli space of Bridgeland stable objects in the Kuznetsov component of
$X_{10}'$. As a result, we show that the Kuznetsov's Fano threefold conjecture
is not true
