The genus conjecture for the prefix-reversal graph on four symbols

Let $\mathbb{P}_4$ be the prefix-reversal graph on the permutations of $\{1,2,3,4\}$, and let $\gamma(\mathbb{P}_4)$ denote its orientable genus. A graph is a pretzel graph if it admits the corresponding pretzel-graph embedding. Pretzel-graph conjecture. The graph $\mathbb{P}_4$ is a pretzel graph, and

$$\gamma(\mathbb{P}_4)=3.$$

The authors report that the displayed rotation system has eight regions and indicate that this is likely the maximum number of regions obtainable from any rotation system, but they did not attempt all possible vertex labelings. Thus the claimed exact genus remains open in the supplied source.