The general Grassmannian reconstruction conjecture

Let $1\leq r<n$ and let $f_E:\operatorname{Gr}(n,r)\rightarrow \mathcal{M}(E)$ be the morphism obtained from the tilting-bundle construction.

General Grassmannian reconstruction conjecture. For any $1\leq r<n$, the morphism

$$f_E:\operatorname{Gr}(n,r)\rightarrow \mathcal{M}(E)$$

is an isomorphism.

The result is proved in the paper for $r=2$. In general, the tilting quiver and a surjective homomorphism to the endomorphism algebra are known, but an explicit system of relations and a compatible Pieri system remain to be constructed, making the general case combinatorially challenging.