On Huisman's conjectures about unramified real curves

Let $X\subset {\mathbb{P}}^n$ be an unramified real curve with $X(\mathbb{R})\mathrm{\ne }\mathrm{\varnothing }$. If $n\ge 3$ is odd, Huisman conjectures that $X$ is an $M$-curve and that every branch of $X(\mathbb{R})$ is a pseudo-line. If $n\ge 4$ is even, he conjectures that $X$ is a rational normal curve or a twisted form of a such. We disprove the first conjecture by giving a family of counterexamples. We remark that the second conjecture follows for generic curves of odd degree from the formula enumerating the number of complex inflection points.