  The Lichtenbaum--Quillen conjecture (LQC) relates special values of zeta
functions to algebraic K-theory groups. The Ausoni--Rognes red-shift
conjectures generalize the LQC to higher chromatic heights in a precise sense.
In this paper, we propose an alternate generalization of the LQC to higher
chromatic heights and give evidence for it at height two. In particular, if the
$n$-th Greek letter family is detected by a commutative ring spectrum $R$, then
we conjecture that the $n+1$-st Greek letter family will be detected by the
algebraic K-theory of $R$. We prove this in the case $n=1$ for
$R=\text{K}(\mathbb{F}_q)$ modulo $(p,v_1)$ where $p\ge 5$ and $q=\ell^k$ is a
prime power generator of the units in $\mathbb{Z}/p^2\mathbb{Z}$. In
particular, we prove that the commutative ring spectrum
$\text{K}(\text{K}(\mathbb{F}_q))$ detects the part of the $p$-primary
$\beta$-family that survives mod $(p,v_1)$. The method of proof also implies
that these $\beta$ elements are detected in iterated algebraic K-theory of the
integers. Consequently, one may relate iterated algebraic K-theory groups of
the integers to integral modular forms satisfying certain congruences.
