Quantized generalized Causality for locally covariant quantum field theories

Let ${*}\text{-}\mathfrak{Alg}$ be the category of non-commutative, associative, $*$-algebras, and let $\mathfrak{Bkgr}_c$ and $\mathfrak{SpBkgr}_c$ be the categories of spacetime manifolds with admissible background fields, respectively with and without globally hyperbolic spacelike cone bundles. Let $\hat{\mathcal{F}}_H\colon\mathfrak{Bkgr}_c\to {*}\text{-}\mathfrak{Alg}$ be a covariant functor and suppose there is a functor $\hat{\mathcal{F}}\colon\mathfrak{SpBkgr}_c\to {*}\text{-}\mathfrak{Alg}$ fitting into the stated commutative diagram and satisfying

$$\mathcal{F}_H(M,\mathcal{B})=\varprojlim \mathscr{F}(M,\mathcal{B}),$$

where the limit is indexed by $\mathfrak{SpBkgr}_c(M,\mathcal{B})\subseteq\mathfrak{SpBkgr}_c$, whose objects are $\mathcal{M}=(M,C^\first,\mathcal{B})$. Quantized generalized Causality conjecture. Whenever the specified Cauchy-pushout diagram exists in $\mathfrak{SpBkgr}_c$, the corresponding diagram with arrows induced by $\hat{\mathcal{F}}$ exists in ${*}\text{-}\mathfrak{Alg}$, with the spacelike separated disjoint union replaced by the independent subsystems tensor product $\hat{\mathcal{F}}(\mathcal{M}_1)\otimes\hat{\mathcal{F}}(\mathcal{M}_2)$. This proposes that the generalized classical Causality construction survives perturbative quantization; no resolution is supplied.