Weak property $(\mathrm{T}_{L^p})$ for discrete groups

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. The key element in the proof is to show that the closedness of the group of $\mathbb T$-valued $1$-coboundaries is a sufficient criterion for strong ergodicity of ergodic p.m.p. actions.