Cohomological conjecture for Schur bundles on Debarre–Voisin hyperkähler manifolds

Let $X\subseteq \operatorname{Gr}(6,10)$ be a Debarre–Voisin hyperkähler manifold, and let $\lambda=(m,t,s,0)$ with $m\geq t+s$. Consider the Littlewood–Richardson decomposition of

$$\Sigma_\lambda\mathcal{Q}\otimes\Sigma_\lambda\mathcal{Q}^{\vee}.$$

Cohomological conjecture. Among its factors, the vector bundle

$$\Sigma_{(2,2,0,0)}\mathcal{Q}\otimes\mathcal{O}_X(-1)$$

is the only one giving a non-trivial contribution to $\operatorname{ext}^1(\Sigma_\lambda\mathcal{Q},\Sigma_\lambda\mathcal{Q})$. Consequently,

$$\operatorname{ext}^1(\Sigma_\lambda\mathcal{Q},\Sigma_\lambda\mathcal{Q})\in\{0,20,40\}.$$

Moreover, among the same factors, $\mathcal{O}_X$ is the only one giving a non-trivial contribution to $\operatorname{ext}^0(\Sigma_\lambda\mathcal{Q},\Sigma_\lambda\mathcal{Q})$; consequently, $\Sigma_\lambda\mathcal{Q}$ is simple.

This conjecture predicts that the displayed Schur bundle controls all nontrivial first self-extensions, while the structure sheaf accounts for the zeroth self-extension. The stated consequences are based on the paper’s Littlewood–Richardson computation; the parser supplies no evidence that the conjecture has been resolved.