Find a proof of redshift for TC(k(n)) independent of syntomic cohomology

Develop a proof, not relying on syntomic cohomology, that the topological cyclic homology TC(k(n)) of the connective cover k(n) of the (2p−2)-periodic Morava K-theory K(n) has chromatic height exactly n+1 (i.e., its T(n+1)-localization is nonzero while its T(m)-localizations vanish for all m>n+1).

Background

The paper proves chromatic redshift for algebraic K-theory of Morava K-theory by analyzing TC(k(n)) via syntomic cohomology, a sophisticated tool drawing on recent advances in prismatic theory.

The authors note that, despite substantial effort, no proof of redshift for TC(k(n)) avoiding syntomic cohomology is known, highlighting a methodological gap. An alternative argument could provide broader conceptual insight and potentially apply in contexts where syntomic techniques are unavailable or unwieldy.