The monochromatic cycle interval conjecture for dense 2-colored graphs

Let $G$ be a graph of order $n$, with vertex set $V(G)$ and edge set $E(G)$. A $2$-coloring of $G$ is a partition

$$E(G)=E(R)\cup E(B),$$

where $R$ and $B$ are spanning subgraphs of $G$. Write $C_k\subset R$ or $C_k\subset B$ when the red or blue graph contains a cycle of length $k$.

Monochromatic cycle interval conjecture. If $n\geq4$ and $\delta(G)>3n/4$, then every $2$-coloring of $E(G)$ has, for every $k\in[4,\lceil n/2\rceil]$, either a red $k$-cycle or a blue $k$-cycle:

$$C_k\subset R\text{ or }C_k\subset B\quad\text{for all }k\in[4,\lceil n/2\rceil].$$

This proposes a Ramsey--Turán strengthening of results for complete graphs and very dense graphs: a minimum-degree condition should force monochromatic cycles of every length in a whole interval, rather than merely one cycle length. The source gives no resolution status for the conjecture.