The biconvex-polytope realization conjecture for tropical linear spaces

Let $M$ be a rank-$k$ matroid on $[n]:=\{1,\dots,n\}$. A tropical linear space is a $(k-1)$-dimensional balanced polyhedral complex in tropical projective space, dual to a coherent matroid subdivision of the base polytope $\mathrm{BP}_M$. A biconvex polytope is a polytope of the type considered in this setting, with cells and tropical linear spaces related by tropical and affine isomorphisms. Biconvex-polytope realization conjecture. Every biconvex polytope is isomorphic to a cell of a tropical linear space, where the isomorphism is a tropical and affine isomorphism. This is the converse to the fact that every bounded cell of a tropical linear space is a biconvex polytope. The paper states that the conjecture is proved by constructing a matroid subdivision dual to any biconvex polytope; the formulation accounts for the fact that matroid subdivisions are not preserved under isomorphisms of biconvex polytopes.