Realising every colour distribution sequence with a Gallai colouring

An edge colouring of $K_n$ with $k$ colours is a Gallai $k$-colouring if it does not contain any rainbow triangle. Gyárfás, Pálvölgyi, Patkós and Wales proved that there exists a number $g(k)$ such that $n\ge g(k)$ if and only if for any colour distribution sequence $(e_1,\cdots ,e_k)$ with ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0px}{}{n}{2}\right)$, there exist a Gallai $k$-colouring of $K_n$ with $e_i$ edges having colour $i$. They also showed that $\mathrm{\Omega }(k)=g(k)=O(k^2)$ and posed the problem of determining the exact order of magnitude of $g(k)$. Feffer, Fu and Yan improved both bounds significantly by proving $\mathrm{\Omega }(k^{1.5}\mathrm{/}\log k)=g(k)=O(k^{1.5})$. We resolve this problem by showing $g(k)=\mathrm{\Theta }(k^{1.5}\mathrm{/}(\log k{)}^{0.5})$.