Bruns–Gubeladze tameness conjecture for polytopal algebra homomorphisms

Let $k$ be a field and let $\operatorname{Pol}(k)$ denote the category of polytopal algebras over $k$. A homomorphism is called tame if it is obtained from the identity mapping $k\to k$ by a sequence of free extensions, Minkowski sums, homothetic blow-ups, polytope changes, and compositions. Bruns–Gubeladze's tameness conjecture. Any homomorphism in $\operatorname{Pol}(k)$ is obtained by such a sequence. Moreover, there are normal forms of such sequences. The conjecture aims to describe arbitrary graded $k$-algebra homomorphisms of polytopal rings; the paper gives supporting results concerning joins, multiples, and products of polytopes, but the general assertion remains open.