Connectedness of the approximate-root set

Let $A$ be a regular ring, let $\alpha,\beta\in\operatorname{Sper} A$, and assume that either $A$ is complete or $\sqrt{\langle\alpha,\beta\rangle}=\mathfrak m$. Let $\mathbf Q_\Lambda$ be the approximate roots common to $\alpha$ and $\beta$, and write the relevant standard expansions using monomials $\mathbf Q^{\theta_{ji}}$ and $\mathbf Q^{\epsilon_{j'i}}$. Approximate-root connectedness conjecture. The set $C$ defined by the valuation inequalities and sign conditions in the source is connected; likewise, the set $C'$ defined by the displayed domination inequalities and the two sign conditions is connected. These are explicit proposed reductions of the Connectedness conjecture; the paper does not establish these assertions in general.