Huisman's conjecture for unramified real curves in even-dimensional projective space

Let $X\subset\mathbb{P}^{n}$ be a smooth, geometrically integral, nondegenerate real curve with $X(\mathbb{R})\neq\emptyset$. A real curve is unramified if every real hyperplane $H\subset\mathbb{P}^{n}$ satisfies

$$\operatorname{wt}(H\cdot X)\leq n-1,$$

where the weight is the degree of the difference between $H\cdot X$ and its reduced divisor. A twisted form of a rational normal curve is a real curve whose base change $X_{\mathbb{C}}$ is a rational normal curve. Huisman's conjecture. If $n\geq 4$ is even and $X\subset\mathbb{P}^{n}$ is an unramified real curve, then $X$ is a rational normal curve or a twisted form of a rational normal curve. Huisman’s computations show nonzero real inflection-point counts for certain positive-genus embeddings, while the classification asserted here remains open.