The maximum-degree-two rainbow matching conjecture

Let $f_G(n,m)$ denote the least number of independent $n$-sets in a graph $G$ that guarantees a rainbow independent set of size $m$. Maximum-degree-two conjecture. If $G$ has maximum degree $2$, then

$$f_G(n,n-1)=n-1.$$

Equivalently, every $n-1$ matchings of size $n$ in a graph of maximum degree $2$ have a full rainbow matching. The source identifies this as a special case of the exceptional-line-graph conjecture and notes that the one-cycle case follows from the chordal-graph theorem.