Product-closure of the original coarse Baum-Connes conjecture remains unknown

Determine whether the class of proper metric spaces satisfying the original coarse Baum-Connes conjecture is closed under products; specifically, ascertain whether X × Y satisfies the coarse Baum-Connes conjecture whenever both X and Y satisfy it.

Background

While the paper proves product-closure for the filtered-coefficients version of the conjecture, it highlights that the analogous statement for the original conjecture (with trivial coefficients) is unknown. The authors’ result provides new product examples for the classical conjecture under stronger hypotheses but does not resolve the general product-closure question.

This uncertainty is historically significant as product-closure impacts the scope of geometric applications of the conjecture and is related to prior work involving coronae and coarse homotopy methods.

References

In generally, if X and Y satisfy the coarse Baum-Connes conjecture, it is unknown whether the conjecture is true or not for X\times Y.

— The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces  (2410.11662 - Zhang, 2024) in Section 1 (Introduction), paragraph preceding Corollary \ref{Cor-intro}

However, it is still unknown in general whether the coarse Novikov conjecture is preserved under finite products.

— The Coarse Novikov Conjecture for Finite Products of Fibred Coarsely Embeddable Spaces  (2609.26643 - Guo et al., 22 Sep 2026) in Section 1, Introduction