When to end the fast

If someone is careful not to perform any forbidden work on Shabbos until the Shabbos is over according to the opinion of Rabbeinu Tam, does he also have to follow this opinion when breaking the fast after a fast day?

Dear …!
Shabbos and fast day end at the same time — at the time that the stars appear in the sky. However, on a fast day there is a reason to be more lenient and not to be stringent according to all opinions, as on Shabbos. Most of the communities that are careful to follow the opinion of Rabbeinu Tam, do this only as a stringency (and not as the letter of the law), and thus their members can be more lenient on a fast day. However, members of communities that have accepted Rabbeinu Tam as their primary authority, such as Satmar and Tzanz, need to be stringent even on a fast day and keep the fast until nightfall according to the opinion of Rabbeinu Tam.

The Shulchan Aruch (Orach Chayim, chapter 261)