Flossing-order conjecture for tree graph-associahedra

For each $n$, consider the set of unlabelled isomorphism classes of trees with $n$ nodes. For a tree $T$, let $\mathcal{B}(T)$ be its graphical building set and let $P_{\mathcal{B}(T)}$ be the corresponding graph-associahedron; write $\gamma_{P_{\mathcal{B}(T)}}$ for its $\gamma$-vector. Flossing-order conjecture. There exists a partial order $\prec$ on these tree classes such that $\mathrm{Path}_n$ is the unique $\prec$-minimum element, $K_{1,n-1}$ is the unique $\prec$-maximum element, and

$$T\prec T'\quad\Longrightarrow\quad\gamma_{P_{\mathcal{B}(T)}}\leq\gamma_{P_{\mathcal{B}(T')}}$$

componentwise. The conjecture would give associahedra and stellohedra as the lower and upper bounds for the $\gamma$-, $f$-, and $h$-vectors of tree graph-associahedra, and would support analogous bounds for connected graph-associahedra. The source reports no resolution.