  Conditioned limit laws constitute an important and well developed framework
of extreme value theory that describe a broad range of extremal dependence
forms including asymptotic independence. We explore the assumption of
conditional independence of $X_1$ and $X_2$ given $X_0$ and study its
implication in the limiting distribution of $(X_1,X_2)$ conditionally on $X_0$
being large. We show that under random norming, conditional independence is
always preserved in the conditioned limit law but might fail to do so when the
normalisation does not include the precise value of the random variable in the
conditioning event.
