Characterization of assembly graphs with genus range {0,1}

Let a double occurrence word (DOW) be a word in which every letter occurs exactly twice, and let $[?]$ denote the graph corresponding to a DOW under the construction used in the paper. For a graph $[?]$, write ${\rm gr}([?])$ for its genus range. A word is loop-nested when it is obtained by loop nesting from another word.

Genus-range characterization. Any DOW whose corresponding graph has genus range $\{0,1\}$ is obtained from the word

$$w=12\cdots n12\cdots n$$

for an odd integer $n$ by loop nesting.

This would characterize all DOWs giving genus range $\{0,1\}$, extending the preceding results on loop-nested words and repeat words. The source provides no resolution status for this conjecture.