Past perfect + adverb when

He was a very prominent boxer. He was a fan favorite, and everybody was certain he would win the match. However, rumors were spreading that he broke his leg WHEN he was skiing earlier that month, and that it made him change his fighting style.

Is the bolded part correct? I used the past simple because of "when". Also, could I use the past perfect instead?
Like this: " Rumors were spreading that he had broken his leg when he had been fighting earlier that month and that it had made him change his fighting style"

A profusion of thanks.