  We consider combining two important methods for constructing
quasi-equilibrium initial data for binary black holes: the conformal
thin-sandwich formalism and the puncture method. The former seeks to enforce
stationarity in the conformal three-metric and the latter attempts to avoid
internal boundaries, like minimal surfaces or apparent horizons. We show that
these two methods make partially conflicting requirements on the boundary
conditions that determine the time slices. In particular, it does not seem
possible to construct slices that are quasi-stationary and avoid physical
singularities and simultaneously are connected by an everywhere positive lapse
function, a condition which must obtain if internal boundaries are to be
avoided. Some relaxation of these conflicting requirements may yield a soluble
system, but some of the advantages that were sought in combining these
approaches will be lost.
