The Strong Connectedness conjecture for regular rings

Let $\Sigma$ be a regular ring. For points $\alpha,\beta\in\operatorname{Sper}\Sigma$ having a common specialization, the Strong Connectedness Property requires that, for elements $g_1,\dots,g_s\in\Sigma\setminus(\mathfrak p_\alpha\cup\mathfrak p_\beta)$ satisfying the valuation inequalities and sign condition in the source, $\alpha$ and $\beta$ lie in the same connected component of

$$\operatorname{Sper}\Sigma\setminus\{g_1\cdots g_s=0\}.$$

More precisely, each $g_i$ satisfies $\nu_\alpha(g_i)\leq\nu_\alpha(\langle\alpha,\beta\rangle)$ and $\nu_\beta(g_i)\leq\nu_\beta(\langle\alpha,\beta\rangle)$, and no $g_i$ changes sign between $\alpha$ and $\beta$.

Strong Connectedness conjecture. A regular ring $\Sigma$ has the Strong Connectedness Property at every pair of points $\alpha,\beta\in\operatorname{Sper}\Sigma$ having a common specialization.

This strengthens the ordinary connectedness framework by weakening the hypotheses on the elements to be avoided. The paper proves the strong conjecture in dimension two and uses it to obtain further cases of the Connectedness and Pierce–Birkhoff conjectures; the general statement remains open.