The generalized partnership and duality conjecture for Gushel–Mukai categories

Let $X_1$ and $X_2$ be Gushel–Mukai varieties, and let $\mathsf{A}(X_1)$ and $\mathsf{A}(X_2)$ be their associated Lagrangian subspaces. Assume that neither subspace contains decomposable vectors. Let $\mathcal{A}_{X_1}$ and $\mathcal{A}_{X_2}$ denote the corresponding GM categories. Generalized partners are defined by an isomorphism identifying the associated Lagrangian subspaces, with the dimensions congruent modulo $2$; generalized duals are defined by an isomorphism identifying one associated Lagrangian subspace with the orthogonal complement of the other, again with dimensions congruent modulo $2$. Generalized partnership and duality conjecture. If $X_1$ and $X_2$ are generalized partners, then there is an equivalence

$$\mathcal{A}_{X_1} \simeq \mathcal{A}_{X_2}.$$

If $X_1$ and $X_2$ are generalized duals, then there is an equivalence

$$\mathcal{A}_{X_1} \simeq \mathcal{A}_{X_2}.$$

The conjecture extends the known birationality results for period partners and dual Gushel–Mukai varieties to generalized partners and generalized duals, predicting that their GM categories are equivalent even when the varieties need not have the same dimension. The supplied source does not establish a resolution of the conjecture.