Polytope realization of transmuted arbor data

Construct, for every arbor t, a simple polytope Φ_t with a distinguished vertex whose F-triangle transforms into the transmutation of the M-triangle of P_t, thereby realizing the corresponding h-vector and polytope data.

Background

The paper formulates a weaker unresolved goal than the full Cambrian-like construction: produce only a simple polytope with a distinguished vertex and the required transformed F- and M-triangle relationship. The authors state that there is no known idea for accomplishing this in general, while proposing halohedra and Hochschild polytopes as partial families.