The conjecture that every non-inane shape is a cube

Let $\Shafinnb$ be the category of shapes, and let

$$\Tower\colon \op{(\NN)}\to\op{(\Shafinnb)}$$

be the functor sending $n$ to $\cubeinc{n}$. Two objects of $\Shafinnb$ are understood to be equivalent when they are isomorphic in its relevant homotopy category. Cube-equivalence conjecture. The functor $\Tower$ is an equivalence of categories. Equivalently, every shape in $\Shafinnb$ is equivalent to $\cubeinc{n}$ for some $n\in\NN$. The preceding results establish this for finite, non-inane shapes whose codomain is a cube; the conjecture asserts that all shapes in $\Shafinnb$ arise in this way up to equivalence.