Smoothness conjecture for Kuranishi spaces of Schur functor bundles

Let $X\subset\operatorname{Gr}(2,V_6)$ be the Fano variety of lines of a smooth cubic fourfold. For a partition $\lambda=(\lambda_1,\lambda_2,\lambda_3,\lambda_4)$, let $\Sigma_\lambda\mathcal{Q}$ be the associated vector bundle on $X$. Kuranishi smoothness conjecture. The Kuranishi space of infinitesimal deformations of $\Sigma_\lambda\mathcal{Q}$,

$$\operatorname{Def}(\Sigma_\lambda\mathcal{Q}),$$

is smooth.

This concerns unobstructedness of deformations: although the obstruction space lies in $\operatorname{Ext}^2(\Sigma_\lambda\mathcal{Q},\Sigma_\lambda\mathcal{Q})$ and can grow with $\lambda$, the conjecture predicts that the Kuranishi space is nevertheless smooth. No general proof is given.