Syzygy behavior in the base locus decomposition program

Let $U$ be a sheaf on the relevant moduli space, and let $W$ be a wall of its stable base locus decomposition. A sheaf is called $F$-admissible when its syzygies arise from the Bridgeland destabilizing object $F$ associated with that wall.

Syzygy behavior conjecture. A sheaf $U$ is $F$-admissible, where $F$ is a Bridgeland destabilizing object that yields the wall $W$ when $U$ enters the stable base locus at the wall $W$.

This statement is presented as the syzygy behavior suggested by the authors' analysis of how minimal free resolutions and interpolation problems determine walls in stable base locus decompositions. The supplied text does not establish its general validity or provide evidence resolving it.