The exceptional-line-graph 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 $m$. Exceptional-line-graph conjecture. If $G$ is the line graph of a graph and is not a graph as in Example, then

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

The source explains that the excluded construction is an obstruction to the stronger bipartite statement and that this conjecture would imply the general-graph full-matching conjecture.