The irreducible hypersurface Pythagoras-number conjecture

Let $f(x,y,z)\in\mathbb{R}[x,y,z]$ be an irreducible polynomial. The irreducible hypersurface Pythagoras-number conjecture. concerns the Pythagoras number $p(A)$ of the coordinate ring $A=\mathbb{R}[x,y,z]/(f(x,y,z))$, namely the least number of squares needed to represent every sum of squares in $A$, if such a finite number exists. The polynomial $f$ is indefinite when it changes sign on $\mathbb{R}^3$; equivalently, the ideal $(f(x,y,z))$ is real.

Irreducible hypersurface Pythagoras-number conjecture. The following conditions are equivalent:

$$p\left(\mathbb{R}[x,y,z]/(f(x,y,z))\right)=+\infty$$

and $f(x,y,z)$ is indefinite.

This would classify the irreducible hypersurface coordinate rings in three variables according to whether their Pythagoras number is finite or infinite. The surrounding discussion presents this as an expectation based on the cases established in the paper; no resolution is supplied here.