  Possible formation scenarios of supermassive black holes in the early
universe include rapid growth from less massive seed black holes (BHs) via
super-Eddington accretion or runaway mergers, yet both of these scenarios would
require seed BHs to efficiently sink to and be trapped in the galactic center
via dynamical friction. This may not be true for their complicated dynamics in
clumpy high-$z$ galaxies. In this work we study this "sinking problem" with
state-of-the-art high-resolution cosmological simulations, combined with both
direct $N$-body integration of seed BH trajectories and post-processing of
randomly generated test particles with a newly developed dynamical friction
estimator. We find that seed BHs less massive than $10^8\,M_\odot$ (i.e., all
but the already-supermassive seeds) cannot efficiently sink in typical high-$z$
galaxies. We also discuss two possible solutions: dramatically increasing the
number of seeds such that one seed can end up trapped in the galactic center by
chance, or seed BHs being embedded in dense structures (e.g. star clusters)
with effective masses above the mass threshold. We discuss the limitations of
both solutions.
