Cheng, Sun, Tan and Wang's rainbow Hamilton cycle conjecture

Let $G$ be a strongly edge-colored graph on $n$ vertices, and let $\delta(G)$ denote its minimum degree. Cheng, Sun, Tan and Wang's conjecture. If

$$\delta(G)\geq\frac{n+1}{2},$$

then $G$ has a rainbow Hamilton cycle. Cheng, Sun, Tan and Wang showed that this minimum-degree condition would be optimal if the conjecture holds; the supplied text gives no resolution.