The bigstar classification-theoretic property for fusions

Let $T$ be a theory, and for each formula $\delta(x,y)$ and model $\mathscr{M}\models T$, write $M^x$ and $M^y$ for the corresponding Cartesian powers and consider the set system

$$\left(M^x;\{\delta(M^x,b):b\in M^y\}\right).$$

Bigstar property conjecture. There is a classification-theoretic property $\bigstar$ such that: (1) there is a property $P$ of set systems for which $T$ has $\bigstar$ if and only if the displayed set system has $P$ for every formula $\delta(x,y)$ and every $\mathscr{M}\models T$; (2) both $\mathrm{NSOP}_1$ and $\mathrm{NTP}_2$ imply $\bigstar$; (3) $\mathrm{NSOP}_1$ is equivalent to the conjunction of $\bigstar$ and $\mathrm{NSOP}$; (4) $\bigstar$ is preserved by fusions under reasonably general conditions; and (5) every fusion of $\mathrm{NIP}$ theories over a stable base has $\bigstar$. The conjecture proposes an as-yet-undiscovered classification-theoretic property that would organize the behavior of fusions and relate existing dividing lines; the source gives no definition of $\bigstar$ or resolution of the proposal.