Abstract: The Lichtenbaum--Quillen conjecture originated as a statement relating algebraic K-theory to étale cohomology and special values of the Riemann zeta function. Waldhausen later reformulated this conjecture as a telescopic homotopy problem. The chromatic redshift problem introduced by Rognes further generalized this idea to higher-height ring spectra. A key tool for understanding the algebraic K-theory of connective ring spectra is the trace method, which we will briefly recall. Work by Hahn--Wilson and others shows that the chromatic redshift problem can be reduced to the Segal conjecture and (weak) canonical vanishing problem. We review the techniques used to prove the Segal conjecture and present examples where it holds. Finally, we demonstrate that the Segal conjecture fails for truncated polynomial algebras over higher-height local number rings, which implies the failure of the Lichtenbaum--Quillen property as well.