  We prove $L^p$-mass concentration properties of Laplace eigenfunctions away
from their nodal sets, extending a recent result in \cite{GM3} to all
dimensions, and giving a slight refinement of a result in \cite{JN}. As a
consequence, we are able to derive a sharp Wasserstein uncertainty principle
that holds uniformly in the high frequency regime, proving a conjecture in
\cite{St1}.
