Ausoni–Rognes redshift conjecture for algebraic K-theory

A $p$-complete, bounded below spectrum $X$ is of fp-type $n$ if the thick subcategory of $p$-local finite complexes $F$ such that $|\pi_*(F\otimes X)|<\infty$ is generated by a type $(n+1)$ complex, meaning a complex with a $v_{n+1}$ self-map. Let $R$ be a suitable $\mathbb{E}_1$-ring of fp-type $n$.

Ausoni–Rognes redshift conjecture. The $p$-completed algebraic K-theory spectrum

$$\mathrm{K}(R)^{\wedge}_p$$

is of fp-type $n+1$.

This is a precise form of the chromatic redshift philosophy, predicting that algebraic K-theory raises chromatic height by exactly one. The statement also implies that $\mathrm{K}(R)\to L^f_{n+1}\mathrm{K}(R)$ is a $p$-local equivalence in sufficiently large degrees. Redshift is known in several height-one examples, but the conjecture is not established for all suitable $\mathbb{E}_1$-rings.