Dubious Loyalties Condition House Bolton

Attached character gains the House Bolton trait.

Response: After you win a challenge in which attached character participated take control of attached character if it is not under your control.

Say I play Dubious Loyalties on an opponent''s character and win a challenge in which that character participates. The character is now under my control. If attachment hate discards my attachment, I continue to control the character, right► In fact, other than an a separate control effect resolving, there is no way for that character to return to the opponent''s control, correct►

Correct. The response has been fully resolved by the time the attachment is discarded in this situation, so you would retain control of the character.

The response has been fully resolved

I´m not sure about this one. I think the response creates a lasting effect (which never resolves fully - it´s indefinitely). In any other case the control would end if the attachment leaves play:

From the FAQ:

(3.26) Duration of Control Change
Unless otherwise stated (for example, with a specified duration), the change of control is permanent until the card that switched sides leaves play or control of the card switches again via a card effect.

Note that attachments that grant control effects end when the attachment granting control leaves play..

The Response that crates the lasting effect has fully resolved. Otherwise, the lasting effect would not exist.

It is also important to realize here that it is not the attachment that is granting the control effect. It is the triggered effect (which creates the lasting effect). It doesn't matter what happens to the card that the lasting effect comes from once the lasting effect is created.

The last part of that FAQ entry is for attachments that say "take control of attached character" as their own, continuous attachment text.