The arbor polytope h-vector realization conjecture

Let $t$ be an arbor, and let $h_t$ denote its $h$-vector. A simple polytope is a polytope whose every vertex lies on exactly as many facets as its dimension; write $h(\Phi_t)$ for the $h$-vector of a simple polytope $\Phi_t$.

The arbor polytope h-vector realization conjecture. For every arbor $t$, there exists a simple polytope $\Phi_t$ such that

$$h_t=h(\Phi_t).$$

This is the weakest of a sequence of proposed realizations relating arbor posets and polytopes. The source says that stronger constructions are not known in general and supplies no resolution of this h-vector realization claim.