Feasible indexing conjecture for successor and ancestor sets of weakly connected DAGs

Let $G=(V,E)$ be a weakly connected directed acyclic graph with $n$ nodes, and let an indexing assign the nodes the labels $1,\ldots,n$. For each node $v$, let $\mathcal S(v)$ be its successor set and let $\mathcal A(v)$ be its ancestor set:

$$\mathcal A(v)=\\{u\in V\backslash\\{v\\}\mid \text{node }u\text{ can reach node }v\\}.$$

Let $LO(\mathcal S(v))$ and $LO(\mathcal A(v))$ denote the associated lexicographic orders, and suppose the indexing satisfies the paper's constraint $LO(\mathcal S(v))\le LO(\mathcal S(v+1))$ for all $v\in[n-1]$.

Feasible indexing conjecture. There exists an indexing satisfying this successor-set ordering and, whenever two consecutive nodes have equal successor-set order,

$$LO(\mathcal S(v))=LO(\mathcal S(v+1))\Rightarrow LO(\mathcal A(v))\le LO(\mathcal A(v+1)),\qquad \forall v\in[n-1].$$

This strengthens the successor-set ordering to break ties using ancestor sets. The paper reports numerical verification for every weakly connected DAG with at most $7$ nodes, but gives no general construction of a feasible indexing; the conjecture remains open.