Exact orientation order of higher real K-theory

Prove or determine whether the complex orientation order of $EO_{\Gamma}$ for a formal group of height $(p-1)k$ is exactly $p^k$, rather than merely known to divide $p^{p^k-1}$.

Background

The complex orientation order Θ(R)\Theta(R) is defined as the least positive integer n such that the n-fold sum of the universal line bundle is R-orientable. The paper explains that existing results provide an upper divisibility bound for the orientation order of higher real K-theories, while a cited conjecture predicts the sharper exact value pkp^k. This unresolved calculation would clarify the orientability input used in constructions of nontrivial stably trivial vector bundles.

References

The orientation order of $EO_{\Gamma}$ has been studied extensively by Bhattacharya--Chatham: $\Theta(EO_{\Gamma})$ divides $p{pk-1}$ when the height of the formal group $\Gamma$ is $n = (p-1)k$, and conjecturally $\Theta(EO_{\Gamma}) = pk$ Main Theorem 1.6 and Conjecture 1.13.

— 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 Remark 3.4, Section 3.2, “Higher real K-theories”