Plesník's chromatic decomposition conjecture for complete graphs

A $k$-decomposition of the complete graph $K_n$ is a decomposition into $k$ spanning subgraphs $G_1,\ldots,G_k$, and for a graph parameter $p$ let

$$p(k;K_n)=\max\left\{\sum_{j=1}^{k}p(G_j)\mid (G_1,\ldots,G_k)\text{ is a $k$-decomposition of }K_n\right\}.$$

Here $\chi$ denotes chromatic number and $\omega$ clique number. Plesník's conjecture. If $k$ and $n$ are positive integers, then

$$\chi(k;K_n)\leq n+{k\choose 2}.$$

This would extend the known equality $\chi(k;K_n)=\omega(k;K_n)$ for $k\leq 3$ and, together with the corresponding clique-number bound, would give the best possible upper bound for convex combinations of $\omega$ and $\chi$.