Conjecture on maximal-length strong exceptional collections for Fano toric Deligne–Mumford stacks

Let $\mathbb{P}_{\mathbf{\Sigma}}$ be a Fano toric Deligne–Mumford stack. A strong exceptional collection of line bundles is a strong exceptional collection whose objects are line bundles, and it has maximal length when its length equals the rank of the Grothendieck $K$-theory group.

Maximal-length fullness conjecture. Every strong exceptional collection of line bundles of maximal length on a Fano toric Deligne–Mumford stack is a full strong exceptional collection.

If true, this would rule out phantom categories as orthogonal complements to maximal-length strong exceptional collections of line bundles on Fano toric Deligne–Mumford stacks. The statement is known in several cases, including smooth toric Fano Deligne–Mumford stacks of Picard number at most two and of arbitrary Picard number in dimension two, but is open in general.