  Possible formation scenarios of supermassive black holes (SMBHs) 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 (DF). 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 DF
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: forming a huge number of
seeds such that one can end up trapped in the galactic center by chance, or
seed BHs being embedded in giant structures (e.g. star clusters) with huge
effective masses above the mass threshold. We discuss the limitations of both
solutions.
