Quenched limit for diffusive biased random walks in random environment

We prove that every directionally transient random walk in random i.i.d.\ environment, under condition (T)γ, which admits an annealed functional limit towards Brownian motion also admits the corresponding quenched limit in d ≥ 2. We exploit a classical strategy that was introduced by Bolthausen and Sznitman but, with respect to the existing literature, we get almost-optimal bounds on the variance of the quenched expectation of certain functionals of the random walk.