Definitions.
Let $G$ be a locally compact group.
Given a unitary representation $(H, \sigma)$ of $G$ on a Hilbert space $H$, we say that a vector $v$ of norm $1$ in $H$ is $(K, \varepsilon)$-invariant, where $K\subseteq G$ is compact and $\varepsilon>0$, if $$\| \sigma(x)v-v \|<\varepsilon$$ for all $x\in K$.
We say that a unitary representation $(H, \sigma)$ has almost invariant vectors if for each $\varepsilon>0$ and each $K\subseteq G$ compact there exists a $(K, \varepsilon)$-invariant unit vector in $H$.
We say that $G$ has property $T_1$ if there exist $\varepsilon>0$ and a compact subset $K$ of $G$ such that for every non-trivial irreducible unitary representation $(H,\rho)$ of $G$ no unit vector in $H$ is $(K, \varepsilon)$-invariant.
We say that $G$ has property $T_2$ if whenever a unitary representation of $G$ has almost invariant vectors, it also has a nonzero invariant vector.
Question.
On page no. 20 of Lubotzky's Discrete Groups, Expanding Graphs and Invariant Measures it is stated that a locally compact group $G$ has property $T_1$ if and only if it has property $T_2$. The author does not provide a proof but gives a few references from where one can read a proof. Two of the references were accessible to me (namely Zimmer's Ergodic Theory and Semisimple Groups and Wang's paper The dual space of semi-simple Lie groups (Theorem 1.5)).
Neither of the two references seem to address the question (however, it is almost surely because of my lack of knowledge in the subject that I was unable to recognize a solution).
So in short, can somebody please help me with the proof of the equivalence of property $T_1$ and property $T_2$.
PS. Please note that the terms 'property $T_1$ and $T_2$' are not standard and are used only for the purpose of this question.