  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 and irreducible 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 an irreducible component of a 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
