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I think the question really is coming from the thought that Strike's
text doesn't indicate 'the controller of this vampire may burn this
card to do x', only:
"Burn to bleed at +1 bleed as a (D) action."
Taking control of the minion includes taking control of all the minion
cards on that minion (according to this LSJ post:
http://groups.google.com/group/rec.games.trading-cards.jyhad/msg/02c1c87c1624d38b
)
However, Conviction cards specifically aren't minion cards, so does
control of a Conviction card on a minion also change when the control
of the minion changes?
-John Flournoy