Factorization homology with adjoints for solidly framed stratified manifolds

Let $n\geq 0$. Write $\operatorname{{\sf Cat}}_n^{\operatorname{{\sf adj}}}$ for the full $\infty$-subcategory of $(\infty,n)$-categories in which every $k$-morphism, for $0<k<n$, has both a left and a right adjoint. Let $(\operatorname{{\sf Cat}}_n^{\operatorname{{\sf adj}}})^{\ast/}$ denote pointed $(\infty,n)$-categories with adjoints, and let $\operatorname{{\mathcal M}\mathsf{fd}}_n^{\operatorname{\sf sfr}}$ be the $\infty$-category of solidly $n$-framed stratified manifolds. Factorization homology with adjoints conjecture. There exists a fully faithful functor

$$\int\colon (\operatorname{{\sf Cat}}_n^{\operatorname{\sf adj}})^{\ast/}\hookrightarrow\operatorname{{\sf Fun}}(\operatorname{{\mathcal M}\mathsf{fd}}_n^{\operatorname{\sf sfr}},\operatorname{\mathcal S\mathsf{paces}})$$

from pointed $(\infty,n)$-categories with adjoints to space-valued functors on solidly $n$-framed stratified manifolds, such that the diagram comparing it with ordinary factorization homology on vari-framed stratified manifolds commutes canonically. This conjecture proposes the factorization-homology theory needed for $(\infty,n)$-categories with adjoints; the paper states it as an assumption and indicates that a proof was expected in forthcoming work.