Sums of squares lengths in coordinates rings of plane curves

Question

let $C\subseteq\mathbb{P}^2$ be a plane algebraic curve and let $\mathbb{R}[C]$ be its real coordinate ring. Let $d\geq 1$ and let $p(d)$ be the smallest number such that every sum of squares in $\mathbb{R}[C]_{2d}$ is a sum of $p(d)$ squares.

Is $p(d)$ known for any $d>>deg(C)$ ? I only know results in the case $\mathbb{R}[x,y,z]$?

Thx