Monochromatic cycle partitions of $2$-coloured graphs with minimum degree $3n\mathrm{/}4$

Balogh, Bar'at, Gerbner, Gy'arf'as, and S'ark"ozy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n\mathrm{/}4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove that this conjecture holds for $n$ large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.