Connectedness conjecture for regular rings

Let $\Sigma$ be a ring and let $\alpha,\beta\in\operatorname{Sper}\Sigma$. The connectedness property at $\alpha$ and $\beta$ requires that, for every finite collection $f_1,\ldots,f_s\in\Sigma\setminus\langle\alpha,\beta\rangle$, there is a connected set $C\subset\operatorname{Sper}\Sigma$ containing $\alpha$ and $\beta$ and avoiding every zero set $\{f_i=0\}$. Let $\mathcal X$ be the class of all regular rings. Connectedness conjecture for regular rings. $CP(\mathcal X)$ holds. This conjecture is a local real-spectrum formulation related to the Pierce–Birkhoff conjecture; its case $s=1$ is the separation conjecture.