Maximum genus range of assembly graphs with an even number of vertices

For an assembly graph with $n$ vertices, let its genus range be the set of genera of its associated ribbon graphs. Let $[a,b]$ denote the integer interval $\{a,a+1,\ldots,b\}$.

Maximum-genus-range conjecture. The maximum genus range of assembly graphs with $2n$ vertices is

$$[n-1,n]$$

for any $n\in\mathbb N$.

The preceding theorem determines the genus range of tangled cords, and the source states this even-vertex assertion as a conjecture. No resolution evidence is supplied.