The adjoint-category functor conjecture

Let ${\mathsf{Vect}}_{\operatorname{inj}}$ be the category of finite-dimensional real vector spaces and injective linear maps, and let ${\mathsf{AdjCat}}_n$ denote the $(\infty,1)$-category of $(\infty,n)$-categories with adjoints. For each inclusion $\mathbb{R}^i\longrightarrow\mathbb{R}^n$ as the first $i$ coordinates, let $\iota_i$ forget all non-invertible morphisms of dimension greater than $i$. The adjoint-category functor conjecture. There is a functor

$${\mathsf{AdjCat}}_{\bullet}\colon {\mathsf{Vect}}_{\operatorname{inj}}^{\operatorname{op}}\longrightarrow \operatorname{\mathscr{C}at}$$

sending $\mathbb{R}^n$ to ${\mathsf{AdjCat}}_n$ and the inclusion $\mathbb{R}^i\longrightarrow\mathbb{R}^n$ to $\iota_i\colon {\mathsf{AdjCat}}_n\longrightarrow {\mathsf{AdjCat}}_i$. The conjecture is motivated by the expected graphical calculus for categories with adjoints, and is known in the restriction to dimensions at most two and $(2,2)$-categories, but remains unproved in general.