Pythagoras number conjecture for two-dimensional regular local rings

Let $R$ be a regular local ring of dimension $2$, and let $K$ be its field of fractions. The Pythagoras number conjecture for regular local rings asserts that

$$p(R)=p(K).$$

Here $p(R)$ and $p(K)$ denote the least numbers of squares needed to represent every sum of squares in $R$ and $K$, respectively. Since $p(R)\geq p(K)$, the conjecture is only interesting when $p(K)$ is finite.