Can one find a net of functions $(f_i)$ in the Fourier algebra of ${\rm SL}_n({\bf Z})$ such that the "amplified net" $(f_i \otimes 1)$, with $1$ denoting the constant function on ${\rm SU}(2)$, acts as an approximate identity for the Fourier algebra of ${\rm SL}_n({\bf Z})\times {\rm SU}(2)$?

The answer is positive for $n=2$ (follows from Haagerup's proof that ${\rm SL}_2({\bf Z})$ is "weakly amenable") but negative for all $n\geq 3$ (this is the statement that for such $n$, ${\rm SL}_n({\bf Z})$ does not have AP, proved by V. Lafforgue and M. de la Salle).