Inscribed right-handed trefoil conjecture

Let $\gamma:\mathbb{R}\to\mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with nonvanishing derivative that parameterizes a right-handed trefoil. A polygonal inscribed trefoil is obtained by cyclically connecting finitely many points of the form $\gamma(t_i)$ by line segments. Inscribed right-handed trefoil conjecture. There exist numbers

$$0\leq t_1<t_2<\cdots<t_6<1$$

such that the polygonal path obtained by cyclically connecting $\gamma(t_1),\gamma(t_2),\ldots,\gamma(t_6)$ is also a right-handed trefoil. This conjecture asks whether the handedness of the inscribed trefoil can be prescribed; the preceding result guarantees an inscribed trefoil for analytically parameterized trefoils but gives no control over its handedness.