It is known that higher rank lattices have property (T) and also that lattices on 2-dimensional Euclidean buildings have property (T) provided the thickness $q+1$ of the building is large enough (which is a condition only in type $\tilde{C}_2$ and $\tilde{G}_2$). My question is about the best known bound that guarantees property (T).

Specifically Żuk states in his famous note that $q \ge 3$ is sufficient for type $\tilde{C}_2$ and that $q \ge 4$ is sufficient for type $\tilde{G}_2$. However, that statement is based on calculations performed by Garland and others (Garland's being the most readable) which only give sufficiency for $q \ge 7$ in type $\tilde{C}_2$ and for $q \ge 11$ in type $\tilde{G}_2$. Is there another justification for Żuk's claim? (Am I making a mistake?)

This is not purely hypothetical: Essert constructs lattices on $\tilde{C}_2$ buildings that lie in the region for which Żuk asserts property (T) but Garland's bounds do not guarantee it (as far as I can see).

Are there other references that give better bounds?

To turn all of this into a concrete question, let me ask this: Do all of the groups described by Essert (Section 0.2, Theorem 0.4) have property (T)?