I think it this case it is a mistake. The author meant to write "work conscientiously" which as you said means "to work hard and carefully". They might have only heard the phrase, without understanding the meanings of "contentious" and "conscientious".
If you google both phrases, you will find a few examples of authors who have made the same mistake, but the correct phrase is far more common.
Your example came from a small Australian Rules football club website - how did you find it?