The tameness conjecture for homomorphisms of polytopal algebras

Let $k[P]$ and $k[Q]$ be polytopal algebras, and let $\operatorname{Pol}(k)$ denote the category whose morphisms are homomorphisms of polytopal algebras. The allowed constructions are free extensions, Minkowski sums, homothetic blow-ups, polytope changes, and compositions; an endomorphism is idempotent when its square equals itself.

Tameness conjecture. Any homomorphism in $\operatorname{Pol}(k)$ is obtained by a sequence of these constructions, starting from the identity mapping $k\to k$. Moreover, there are normal forms of such sequences for idempotent endomorphisms.

The conjecture says that the subcategory of homomorphisms obtained in this way, called the tame subcategory, is the whole category $\operatorname{Pol}(k)$. The source gives no resolution, so the conjecture remains open.