Markov degree bound for triangulated spheres

Let $\Gamma$ be a triangulation of a sphere of dimension $n$. The Markov basis of $I_{\Gamma, {\bf 2}}$ is a finite set of binomials associated to the simplicial complex $\Gamma$. Markov degree conjecture for triangulated spheres. Every element of the Markov basis of $I_{\Gamma, {\bf 2}}$ has degree at most $2^{n+1}$. This conjecture is motivated by the bipyramid example and related results for graph decompositions; its general status is not resolved in the supplied source.