  The source coding problem with encoded side information is considered. A
lower bound on the strong converse exponent has been derived by Oohama, but its
tightness has not been clarified. In this paper, we derive a tight strong
converse exponent. For the special case such that the side-information does not
exists, we demonstrate that our tight exponent of the WAK problem reduces to
the known tight expression of that special case while Oohama's lower bound is
strictly loose. The converse part is proved by a judicious use of the
change-of-measure argument, which was introduced by Gu-Effros and further
developed by Tyagi-Watanabe. Interestingly, the soft Markov constraint, which
was introduced by Oohama as a proof technique, is naturally incorporated into
the characterization of the exponent. A technical innovation of this paper is
recognizing that the soft Markov constraint is a part of the exponent, rather
than a penalty term that should be vanished. In fact, via numerical experiment,
we provide evidence that the soft Markov constraint is strictly positive. The
achievability part is derived by a careful analysis of the type argument;
however, unlike the conventional analysis for the achievable rate region, we
need to derive the soft Markov constraint in the analysis of the correct
probability. Furthermore, we present an application of our derivation of strong
converse exponent to the privacy amplification.
