Kazhdan property conjecture for automorphism groups of free groups

Let $F_k$ be the free group of rank $k$, and let $\operatorname{Aut} F_k$ denote its automorphism group. Kazhdan property $(T)$ is the property referred to in the source for a group satisfying Kazhdan's fixed-point condition.

Kazhdan property conjecture. For $k\geqslant 4$, $\operatorname{Aut} F_k$ satisfies Kazhdan property $(T)$.

This is described as a long-standing conjecture and would, through the Lubotzky–Pak observation, imply logarithmic mixing-time bounds for random walks on components of product replacement graphs. Its resolution status is not specified in the supplied text.