Morava K-theory analogue of Yagita’s conjecture

Prove, for every compact connected Lie group K with maximal torus T and complexification G = K_C, that for A equal to the algebraic connective Morava K-theory CK(n) or the algebraic periodic Morava K-theory K(n), the equality A*(G) = T*(A^{top}(K/T)) holds; equivalently, determine whether the natural map A*(G) -> A^{top}(K) is injective for all such K.

Background

The authors propose and partially verify an analogue of Yagita’s conjecture for Morava K-theories, showing it for SO(m) and Spin(m). They formulate Conjecture 1 as an exact equality between algebraic and topological Morava cohomology via the projection K -> K/T.

This conjecture generalizes the comparison between algebraic and topological theories, aiming to characterize A*(G) through the image of the topological cohomology under T*.