4 problems

Indecomposable ACM-bundle conjecture. Each indecomposable ACM bundle of rank four is a $2$-extension bundle, and no indecomposable ACM bundle has rank two or three.

BGS conjecture. aCM bundles of ranks smaller than the minimal rank of an Ulrich bundle on a smooth hyperquadric do not exist on $X$.

Let $Y\subset\mathbb{P}^s$ be a smooth hypersurface, and let an aCM or Ulrich vector bundle on $Y$ have rank $r$. The lower-bound conjecture. The rank should satisfy … This conject…

Buchweitz–Greuel–Schreyer splitting conjecture. If $\operatorname{rank} E<2^e$, then $E$ splits. In particular, the conjecture predicts that every ACM bundle of rank $3$ on a hyper…