Simplicial and flag-simplicial h-vector conjectures for preorder polytopes

Let $\tau$ be a preorder of size $n$. For $0\leq i\leq n$, let $h_i(\tau)$ count the lattice points of $\mathcal Q_\tau$ with exactly $i$ nonzero coordinates, and write $h(\tau)=(h_0(\tau),\ldots,h_n(\tau))$. Simplicial h-vector conjecture. For every preorder $\tau$ of size $n$, the following assertions hold in increasing strength: (a) $h_i(\tau)=h_{n-i}(\tau)$; (b) $h(\tau)$ is palindromic and $h_0(\tau)\leq\cdots\leq h_{\lfloor n/2\rfloor}(\tau)$; (c) $h(\tau)$ is the $h$-vector of an $n$-dimensional simplicial polytope; and (d) $h(\tau)$ is the $h$-vector of an $n$-dimensional flag simplicial polytope. The source presents these as a hierarchy of conjectures and gives no resolution for general preorders.