Kazhdan constants for two-element generating sets of $\mathrm{SL}_n(\mathbb Z)$

For $n\geq3$, the group $\mathrm{SL}_n(\mathbb Z)$ has property $(T)$, so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum