The permutation-representation conjecture for comparability graphs

Let $G$ be a comparability graph, meaning the graph of a partial order. Write $\mathcal{R}_2$ for the class of graphs with representation number at most two, and $\mathcal{R}^p_{\le 3}$ for the class of graphs with permutation-representation number at most three. Permutation-representation conjecture. If

$$G\in\mathcal{R}_2,$$

then

$$G\in\mathcal{R}^p_{\le 3}.$$

This conjecture proposes that every comparability graph with representation number at most two has permutation-representation number at most three. It is presented as an observation-based conjecture in the paper; no resolution is supplied here.