I think it is appropriate to let MO users know (the OP himself knows it well) that this question was recently solved: it is feasible to provide a computer based proof for property (T) using the Ozawa Equation and this method was applied successfully to the groups mentioned in the question. In particular, the famous open problem alluded to above is now solved. The computation is done via semidefinite programming, as suggested by David Speyer in his answer.
It is now known that the group $\mathrm{Aut}(F_n)$ has property (T) iff $n\geq 4$.
For $n\geq 5$ this is shown in Kaluba-Kielak-Nowak. The case $n=5$ was treated earlier in Kaluba-Nowak-Ozawa and the case $n=4$ was recently settled in Nitsche. Some earlier positive results were obtained in Netzer-Thom and Fujiwara-Kabaya. $\mathrm{Aut}(F_2)$ and $\mathrm{Aut}(F_3)$ are known not to have (T). This is all quite remarkable.