Consistency of the Greedy Sparsest Poset algorithm

Let $P$ be the joint distribution of random variables $(X_i)_{i \in V}$, with $V$ finite, and let $G^*$ be a directed maximal ancestral graph (DMAG) on $V$. Assume that $P$ is Markov to $G^*$, that is, every $m$-separation in $G^*$ is a conditional independence of $P$, and that $P$ is restricted-faithful to $G^*$: $X_i$ and $X_j$ are conditionally dependent given $X_S$ for every $S \subseteq V \setminus \{i, j\}$ such that $i$ and $j$ are $m$-connected given $S$ in $G^*$ and are adjacent, or are the endpoints of a path of length two in the skeleton of $G^*$, or are the endpoints of a discriminating path.

For a partial order $\pi$ on $V$ and $i \ne j$, let $$D_\pi(i, j) = \{v : v \le_\pi i \text{ or } v \le_\pi j\} \setminus \{i, j\}.$$ Let $A(\pi, P)$ be the graph on $V$ with an edge between $i$ and $j$ exactly when $X_i$ and $X_j$ are conditionally dependent given $X_{D_\pi(i,j)}$, directed from the smaller to the larger endpoint when they are comparable in $\pi$ and bidirected otherwise. Let $\operatorname{po}(H)$ be the partial order of directed ancestry in $H$, equality included, and set $$G_\pi = \operatorname{cl}_{\mathrm{ma}}\Bigl(A\bigl(\operatorname{po}(A(\pi, P)), P\bigr)\Bigr),$$ where $\operatorname{cl}_{\mathrm{ma}}$ is the maximal ancestral closure, which adds an edge between every pair that cannot be $m$-separated while preserving the separation model and the ancestral orientations. The search graph has as vertices the distinct graphs $G_\pi$, with an arc from $H$ to $G_{\operatorname{po}(H')}$ whenever $H'$ is obtained from $H$ by a legitimate mark change: changing one edge $i \to j$ into $i \leftrightarrow j$, or conversely, so that $H'$ is still a DMAG Markov equivalent to $H$. Write $|E(H)|$ for the number of edges of $H$.

Bernstein, Saeed, Squires and Uhler, who introduced this search as the Greedy Sparsest Poset (GSPo) algorithm, proved that under these assumptions every $G_\pi$ is a minimal independence map of $P$ and every $G_\pi$ with the fewest edges is Markov equivalent to $G^*$. Zhang and Spirtes proved that two DMAGs are Markov equivalent if and only if one can be transformed into the other by a sequence of legitimate mark changes, and characterized the legitimate mark changes graphically.

Problem. Does every vertex $H_0$ of the search graph have a directed path $H_0, H_1, \dots, H_r$ in it with $$|E(H_{t+1})| \le |E(H_t)| \quad (0 \le t < r), \qquad |E(H_r)| = \min_\pi |E(G_\pi)|?$$ This is the oracle consistency of GSPo with a sufficiently large search depth; no fixed depth or polynomial running time is prescribed.