The converse characterization for 1-dimensional bipartite hypergraphs

Let $\mathcal{H}$ be a 1-dimensional bipartite hypergraph with

$$\operatorname{pd}(\mathcal{H})\leq |V(\mathcal{H})|-2.$$

Converse characterization conjecture. Then $\operatorname{pd}(\mathcal{H})=|V(\mathcal{H})|=2$ if and only if $\mathcal{H}$ satisfies $(\sharp)$. The authors are asking whether this gives a combinatorial characterization of 1-dimensional bipartite hypergraphs of projective dimension $|V(\mathcal{H})|-2$; the statement is presented as a potential characterization and no resolution is given.