Barát–Gyárfás–Sárközy conjecture for rainbow matchings in multigraphs

Let $G$ be a properly edge-colored multigraph with $2n-t_n$ colors and exactly $n$ edges of each color, where

$$t_n=\begin{cases}0&\text{if }n\text{ is even},\\1&\text{if }n\text{ is odd}. \end{cases}$$

Barát–Gyárfás–Sárközy conjecture. Then $G$ has a rainbow matching using $n$ colors.

The conjecture concerns the number of colors needed when every color class has exactly $n$ edges. The source cites best-known partial results, but does not state a resolution of the conjecture.