  We show that, for a countable discrete group $\Gamma$, property
$(\mathrm{T}_{L^p})$ of Bader, Furman, Gelander and Monod is equivalent to the
property that, whenever an $L^p$-representation of $\Gamma$ admits a net of
almost invariant unit vectors, it has a non-zero invariant vector. Central in
the proof is to show that the closure of the group of $\mathbb{T}$-valued
$1$-coboundaries is a sufficient criteria for strong ergodicity of ergodic
p.m.p. actions.
