Polynomial surfaces of revolution parametrization conjecture

Let $S$ be a polynomial surface of revolution, and let $p(t)$ be its associated polynomial. A proper real polynomial parametrization is a real polynomial map that parametrizes $S$ properly. PSOR conjecture. The surface $S$ has a proper real polynomial parametrization if and only if it is two-dimensional as a real set and the associated polynomial $p(t)$ has at most one real root.

The conjecture extends the verified cases $\Delta_S=1,2$ and is supported by an explicit example for $\Delta_S=3$. Its verification is stated to remain open.