Pure O-sequence conjecture for PS ear-decomposable simplicial complexes

Let $\Delta$ be a PS ear-decomposable simplicial complex, and let $h(\Delta)$ denote its $h$-vector. The PS ear-decomposability conjecture. The vector $h(\Delta)$ is a pure $\mathcal{O}$-sequence. This conjecture extends Stanley's conjecture from matroid simplicial complexes, since coloop-free matroid simplicial complexes are PS ear-decomposable; the paper studies this claim in dimension two and proves the corresponding result for $2$-dimensional PS ear-decomposable simplicial complexes.