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) _, 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.