  In this short paper we prove the global regularity of solutions to the
Navier-Stokes equations under the assumption that slightly supercritical
quantities are bounded. As a consequence, we prove that if a solution $u$ to
the Navier-Stokes equations blows-up, then certain slightly supercritical
Orlicz norms must become unbounded. This partially answers a conjecture
recently made by Terence Tao. The proof relies on quantitative regularity
estimates at the critical level and transfer of subcritical information on the
initial data to arbitrarily large times. This method is inspired by a recent
paper of Aynur Bulut, where similar results are proved for energy supercritical
nonlinear Schr\"odinger equations.
