The characterization of destabilization sets by stability characters

Let $U$ be a coherent sheaf with a minimal free resolution

$$0\to\bigoplus_{i=1}^r\mathcal{O}_{\mathbb{P}^2}(a_i)\xrightarrow{M}\bigoplus_{i=1}^s\mathcal{O}_{\mathbb{P}^2}(b_i)\to U\to0.$$

Let the destabilization set of $U$ consist of the objects destabilizing $U$ at its Bridgeland wall, and call $U$ stability $\zeta$-admissible when $M$ is a general destabilizing $\zeta$-map.

Characterization conjecture. Every element of the destabilization set of $U$ comes from vanishing conditions on the map $M$. Moreover, knowing the destabilization set of $U$ is equivalent to knowing all characters $\zeta$ for which $U$ is stability $\zeta$-admissible.

This is the stated strengthening of the preceding vanishing-entry conjecture and links the geometry of stable base loci to the combinatorics of minimal free resolutions. No resolution evidence is supplied, so the conjecture is recorded as open.