The Separation conjecture for real spectra

Let $R$ be a real closed field, let $A=R[x_1,\dots,x_n]$, and let $\operatorname{Sper} A$ be the real spectrum of $A$. For $\alpha,\beta\in\operatorname{Sper} A$, write $\langle\alpha,\beta\rangle$ for their separating ideal, and let $g\in A$.

Separation conjecture. If g\text{\in \hspace{-.8em}/}\langle\alpha,\beta\rangle, then $\alpha$ and $\beta$ lie in the same connected component of

$$\operatorname{Sper} A\setminus\{g=0\}.$$

The source describes this as a weaker stepping stone toward the Pierce–Birkhoff conjecture and proves that it implies the Pierce–Birkhoff conjecture for functions defined by two polynomials. No general resolution is given.