Classification conjecture for absolutely cartesian cubes of spaces

Let an $n$-cube of spaces be a diagram indexed by the powerset of $[n]=\{0,1,\ldots,n\}$, and call it absolutely cartesian when every homotopy functor sends it to a homotopy cartesian cube. A map of two $(n-1)$-cubes is an $n$-cube whose two constituent $(n-1)$-dimensional faces are the source and target cubes; cubes of these types can be composed along a shared face. Classification conjecture. An $n$-cube of spaces is absolutely cartesian if and only if it can be written as either a map of two absolutely cartesian $(n-1)$-cubes or a chain of compositions of $n$-cubes of these types. The forward implication is unknown in general; the reverse implication follows from the inductive construction described in the paper, with the square case supplied by the classification theorem proved there.