Assess ease of verifying Kirchberg Factorization Property versus residual finiteness for property (T) discrete quantum groups

Ascertain whether, for discrete quantum groups with property (T), checking the Kirchberg Factorization Property (equivalently, amenability of the Haar state on C(\mathbb G_{max})) is easier in practice than verifying residual finiteness, given that these properties are equivalent in this setting.

Background

For discrete quantum groups with property (T), residual finiteness is known to be equivalent to the Kirchberg Factorization Property, which can be phrased in terms of amenability of the Haar state on the maximal C*-algebra of the compact dual.

Despite this equivalence, it is unclear whether one property is practically simpler to verify than the other. Clarifying this could guide how to establish the existence of explicit quantum expander families via representation-theoretic or C*-algebraic criteria.