Universality conjecture for the universal bulk-boundary system

Let $\mathcal T$ be a Poisson BV theory on $N$, and let $\mathcal Z(\mathcal T)$ be its universal bulk theory. Let

$$\rho:N\times\mathbb R_{\geq 0}\to N,\qquad \tau:N\times\mathbb R_{\geq 0}\to\mathbb R_{\geq 0}$$

be the respective projections. Write $\operatorname{Obs}^{\mathrm{cl}}_{\mathcal Z(\mathcal T),\mathcal T}$ for the factorization algebra of classical observables of the bulk-boundary system $(\mathcal Z(\mathcal T),\mathcal T)$. Universality conjecture. The factorization algebra $\operatorname{Obs}^{\mathrm{cl}}_{\mathcal Z(\mathcal T),\mathcal T}$ is terminal among $\mathbb P_0$-factorization algebras $\mathcal F$ such that: (1) there is an equivalence $\operatorname{Obs}^{\mathrm{cl}}_{\mathcal T}\to\rho_*\mathcal F$ of $\mathbb P_0$-factorization algebras on $N$; and (2) for every $U\subseteq N$, the factorization algebra $\tau_*\left(\left.\mathcal F\right|_{U\times\mathbb R_{\geq 0}}\right)$ on $\mathbb R_{\geq 0}$ is stratified locally constant, meaning that inclusions of the forms $[0,a)\subseteq[0,a')$ and $(a,b)\subseteq(c,d)$ separately induce equivalences. Equivalently, for every $\mathcal F$ satisfying (1) and (2), there exists, up to an appropriate notion of homotopy, a map $\mathcal F\to\operatorname{Obs}^{\mathrm{cl}}_{\mathcal Z(\mathcal T),\mathcal T}$ of factorization algebras on $N\times\mathbb R_{\geq 0}$. This conjecture is intended to justify the term “universal” in “universal bulk-boundary system”; its resolution would characterize the universal bulk-boundary system by a terminal property among systems with the prescribed boundary observables and stratified-local-constant behavior in the normal direction.