Realisation of formal invariant curves for algebraically isolated vector fields

Let $X$ be a smooth germ of vector field at the origin of $\mathbb{R}^{n}$ with an algebraically isolated singularity, and suppose it has a formal invariant curve $\hat{\gamma}\in\hat{\mathcal{E}}_{1}^{n}$. Realisation conjecture. There exists $\gamma\in\mathcal{E}_{1}^{n}$ such that

$$T_{\underline{0}}\gamma=\hat{\gamma}$$

and $\gamma$ is an invariant curve of $X$. The preceding planar theorem establishes the analogous assertion in dimension two; the statement proposes the corresponding general realisation result in arbitrary dimension.