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We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a specialsmooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smoothirreducible projective threefold if $X_{10}$ is general among specialGushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. Weconstruct an irreducible component of a moduli space of Bridgeland stableobjects 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 threefoldwith moduli space of Bridgeland stable objects in the Kuznetsov component of$X_{10}'$. As a result, we show that the Kuznetsov's Fano threefold conjectureis not true

Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture