Near diagonal additive energy bound for points on algebraic surfaces

Let $F:{\mathbb{R}}^3\to \mathbb{R}$ be a polynomial that is irreducible over $\mathbb{R}$ with $\text{deg}(F)\ge 2$. We prove that, for any finite $X\subset Z(F)$ that does not concentrate on affine lines, $$E(X)=\mathrm{\#}\{(a,b,c,d)\in X^4:a+b=c+d\}{\ll }_{\text{deg}(F),\epsilon }(\mathrm{\#}X{)}^{2+\epsilon }\mathrm{.}$$