The crown-subgraph conjecture for representation numbers of bipartite graphs

Let $G$ be a bipartite graph. For $\kappa_0\geq 5$, let $H_{\kappa_0,\kappa_0}$ be a largest induced crown graph in $G$, and let $\mathcal{R}(G)$ denote the representation number of $G$.

Crown-subgraph conjecture. The representation number of $G$ equals either

$$\left\lceil\frac{\kappa_0}{2}\right\rceil$$

or

$$\left\lceil\frac{1+\kappa_0}{2}\right\rceil.$$

This conjecture proposes that the representation number is determined, up to the two displayed possibilities, by the largest induced crown graph in $G$. The source presents it in the conclusion and gives no evidence of a resolution.