Can a combination of the conformal thin-sandwich and puncture methods yield binary black hole solutions in quasiequilibrium?

We consider combining two important methods for constructing quasiequilibrium 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 quasistationary and that avoid physical singularities while simultaneously are connected by an everywhere positive lapse function, a condition which must be obtained 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.