Fujita and Magnant's vertex-pancyclic conjecture for properly colored cycles

Let $G$ be an edge-colored complete graph on $n$ vertices. Write $\delta^c(G)$ for its minimum color degree, where the color degree of a vertex is the number of colors on its incident edges. A cycle is properly colored if consecutive edges have distinct colors.

Fujita and Magnant's conjecture. If

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

then every vertex of $G$ is contained in properly colored cycles of every length from $3$ to $n$.

Fujita and Magnant proved the assertion for properly colored triangles and quadrangles, and later work established it for cycles of length at least the minimum color degree. The conjecture is known when the graph contains no monochromatic triangles, but is otherwise open.