Weak property (TLᵖ) for discrete groups

We show that, for a countable discrete group Γ, property (TLᵖ) of Bader, Furman, Gelander and Monod is equivalent to the property that, whenever an Lᵖ-representation of Γ 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 T-valued $1$-coboundaries is a sufficient criteria for strong ergodicity of ergodic p.m.p. actions.