Buhl cycle-condition converse for the weak maximum likelihood threshold

Let $G$ be a graph with at least one edge. An orientation of $G$ is acyclic if it has no directed cycles, and a cycle is stretched when it has the form $v_1\to v_2\to\dots\to v_k\leftarrow v_1$. Let $\operatorname{wmlt}(G)$ denote the weak maximum likelihood threshold. Buhl cycle-condition converse. If $G$ has an acyclic orientation with no stretched cycles, then

$$\operatorname{wmlt}(G)=2.$$

The forward implication is known: weak maximum likelihood threshold $2$ implies the existence of such an orientation. The conjecture asserts the converse, completing the rigidity-theoretic characterization of graphs with weak maximum likelihood threshold $2$.