The linear-length monochromatic cycle conjecture for dense graphs

Let $G$ be a graph of sufficiently large order $n$, and let $0<c<1$. A $2$-coloring of $E(G)$ assigns each edge of $G$ one of two colors, red or blue. A monochromatic cycle is a cycle all of whose edges receive the same color.

Linear-length monochromatic cycle conjecture. If

$$\delta(G)>cn,$$

then every $2$-coloring of $E(G)$ contains a monochromatic cycle $C_k$ for some $k\geq cn$.

This is presented as a seemingly easier conjecture than the preceding cycle-interval conjecture. It asks whether a positive minimum-degree proportion alone forces a monochromatic cycle whose length is at least the same linear proportion of the order; the source gives no resolution status.