Makeev's inscribed quadrangle conjecture for plane curves

Let $C$ be a smooth simple closed curve in $\mathbb R^2$, and let $Q$ be four points on a single circle. Makeev's inscribed quadrangle conjecture. There is a similarity transform $\sigma$ with positive determinant such that

$$\sigma(Q)\subset C.$$

The source describes this as another conjecture of Makeev and says that some particular cases have been proved positively; the general assertion remains unresolved in the supplied text.