The uniqueness conjecture for chord diagrams with genus range {1,2}

Let $n$ be a positive integer, and let a chord diagram be represented by a double-occurrence word in which each letter occurs exactly twice; chord diagrams and words are considered up to the equivalence used in the paper. The genus range is the set of genera of the thickened chord diagrams associated with the chord diagram.

Uniqueness conjecture for genus range $\{1,2\}$. For any $n\neq 4$, there is a unique, up to equivalence, chord diagram with genus range

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

and it is represented by

$$(123123)(44\cdots nn).$$

The assertion is supported by the computed examples, while the paper gives no general proof. The excluded value $n=4$ is an exceptional case in the stated uniqueness claim.