Geometric connectedness conjecture for Langlands-parameter summands

Let $F$ be the local field, let $p$ be its residue characteristic, let $W_F^0$ be the chosen subgroup of the Weil group, and let $\phi$ and $\alpha$ be a pair indexing a summand. The scheme ${\underline Z}^{1}(W_F^0,{\hat G})_{\phi,\alpha}$ is defined over $\mathcal{O}_{K_e}[1/p]$.

Connectedness conjecture. For any pair $(\phi,\alpha)$, the $\mathcal{O}_{K_e}[1/p]$-scheme ${\underline Z}^{1}(W_F^0,{\hat G})_{\phi,\alpha}$ is connected and remains connected after any finite flat integral base change.

This is presented as the main conjecture over $\bar{\mathbb Z}[1/p]$: the decomposition referred to in the source should be the decomposition into connected components. The source gives no resolution status.