Finiteness of the higher real K-theory Hurewicz image

Determine whether the Hurewicz image in the homotopy groups of the higher real K-theory spectrum $EO_{p-1}$ at primes $p\geq 5$ is infinite or finite.

Background

The paper uses higher real K-theory spectra EOp−1EO_{p-1} to detect torsion in stable homotopy groups and unstable homotopy groups of unitary groups. At primes p≥5p\geq 5, the authors describe a finite family of classes known to lie in the Hurewicz image, while noting that the overall size of this image has not been established and is suspected to be finite.

References

For $p\geq 5$, the Hurewicz image in $\pi_*EO_{p-1}$ is not known to be infinite, and in fact is widely suspected to be finite.

— High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories  (2608.19411 - Chatham et al., 19 Aug 2026) in Section 2, Subsection 2.3, “Higher real K-theories”