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Existence of a pair of discrete groups whose reduced group C*-algebras fail Tomiyama’s property (F)

Identify discrete groups Γ1 and Γ2 such that the pair of reduced group C*-algebras C*_red(Γ1) and C*_red(Γ2) does not satisfy Tomiyama’s property (F).

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Background

Tomiyama’s property (F) is central to ideal separation phenomena in minimal tensor products and, via the authors’ results, governs the ISP for Cartesian products of groups. Finding discrete groups whose reduced C*-algebras fail property (F) would provide counterexamples to broad ideal separation expectations and inform the structure of tensor products.

References

It remains unclear if such a pair of groups exists; see Question 4.16 and the discussion preceding it.

Question 4.16. Are there discrete groups T1 and I2 such that the pair of C *- algebras Cred(T'1) and Cred(T2) does not have Tomiyama's property (F)?

— The ideal separation property for reduced group $C^*$-algebras (2408.14880 - Austad et al., 27 Aug 2024) in Section 4, Question 4.16