An unknown-order-property characterization of quadratic atomicity and binary error

Let $\mathrm{XOP}_2$ be a prospective, as-yet-undefined generalization of the order property, and write $\mathrm{NXOP}_2$ for its negation. Let a property be quadratically atomic in the group setting and admit binary $\operatorname{disc}_{2,3}$-error in the hereditary $3$-uniform-hypergraph setting, using the paper's definitions. XOP2 conjecture. There exists a definition of $\mathrm{XOP}_2$ such that: (i) if an epGP $\mathcal{P}$ is $\mathrm{NXOP}_2$, then $\mathcal{P}$ is quadratically atomic; and (ii) if a hereditary property $\mathcal{H}$ of $3$-uniform hypergraphs is $\mathrm{NXOP}_2$, then $\mathcal{H}$ admits binary $\operatorname{disc}_{2,3}$-error. This is presented as an open question motivated by examples that obstruct quadratic atomicity and binary error; no resolution is supplied.