Tiling edge-coloured graphs with few monochromatic bounded-degree graphs

We prove that for all integers $\Delta,r \geq 2$, there is a constant $C = C(\Delta,r) >0$ such that the following is true for every sequence $\mathcal{F} = \{F_1, F_2, \ldots\}$ of graphs with $v(F_n) = n$ and $\Delta(F_n) \leq \Delta$, for each $n \in \mathbb{N}$. In every $r$-edge-coloured $K_n$, there is a collection of at most $C$ monochromatic copies from $\mathcal{F}$ whose vertex-sets partition $V(K_n)$. This makes progress on a conjecture of Grinshpun and S\'ark\"ozy.