Exact monochromatic circumference conjecture at minimum degree three quarters

Let $G$ be a graph of order $n$ with minimum degree

$$\delta(G)\geq\frac{3}{4}n.$$

Write $n=3t+r$ with $r\in\{0,1,2\}$, and let

$$E(G)=E(R_G)\cup E(B_G)$$

be a $2$-edge colouring. The exact monochromatic circumference conjecture. The graph $G$ has a monochromatic cycle of length at least $2t+r$; equivalently, its monochromatic circumference is at least $2t+r$.

The claim strengthens the asymptotic circumference result at minimum degree at least $\frac{3}{4}n$. The supplied text does not state a proof or resolution, so the conjecture remains open.