Generation conjecture for local face modules of vertex-induced triangulations

Let $\sigma \\: \Gamma \to 2^V$ be a vertex-induced triangulation of a $(d-1)$-dimensional simplex, and let $k$ be a field. Generation conjecture. The local face module $L(\Gamma)$ is generated as a $K[\Gamma]$-module in degree at most $d/2$. The conjecture is motivated by the strong Lefschetz property; the authors report no known vertex-induced examples where this generation property fails.