A note on a sheaf of real regular functions

Coste and Roy defined a structural sheaf on the real Zariski spectrum of a semi-real ring $A$ and asked whether the ring of global sections is $\sum_1^{-1} A$ when $\Sigma_1$ is the multiplicative subset $\{ 1+\sum_{i=1}^{n}a_i^{2}\,|a_i\in A, n\in\mathbb{N}\cup\{0\}\}$ of $A$. We give a positive answer to this question in the case of $A$ is a real ring