The reduced Homological Vanishing Conjecture

Let $p$ be a prime and $h$ a height. Write $\mathbf{G}_h$ for the height-$h$ Morava stabilizer group, $E_h$ for Morava $E$-theory, and let $\mathbf{F}_p\subseteq \mathbf{F}_{p^h}$ denote the prime field. The natural inclusion of Witt vectors induces a map on zeroth homology.

Reduced Homological Vanishing Conjecture. For the parameters under consideration, this map is an isomorphism

$$\mathbf{F}_p\simeq H_0(\mathbf{G}_h;\mathbf{F}_{p^h})\overset{\sim}{\longrightarrow} H_0(\mathbf{G}_h;\pi_0(E_h)/p).$$

This is the reduced homological special case of the Chromatic Vanishing Conjecture. The paper explains that this special case would imply vanishing of the exotic $K(h)$-local Picard group in the relevant borderline situation, but does not establish the conjecture in general.