The uniqueness conjecture for chord diagrams with genus range {0,1}

Let $n$ be a positive integer, and let a double-occurrence word be a word in which each of the letters $1,\ldots,n$ occurs exactly twice. Double-occurrence words are considered up to equivalence, and each such word determines a chord diagram. The genus range of a chord diagram is the set of genera of its thickened chord diagrams.

Uniqueness conjecture for genus range $\{0,1\}$. For any $n\neq 2$, there is a unique, up to equivalence, double-occurrence word

$$11\cdots nn$$

that corresponds to a chord diagram with genus range $\{0,1\}$.

The claim is based on the computed cases in the paper. The exceptional case $n=2$ is explained by the two inequivalent words $1122$ and $1212$, both of which have genus range $\{0,1\}$; no general proof is supplied.