The two-value genus-range conjecture for chord diagrams

Let $D$ be a chord diagram with $n>0$ chords. Its genus range is the set of genera of the thickened chord diagrams associated with $D$.

Two-value genus-range conjecture. If the genus range of $D$ consists of two numbers, then it is either

$$\{0,1\}$$

or

$$\{1,2\}.$$

The paper reports that this holds for all computed examples, but provides no general proof.