Mixed chromatic-clique decomposition bound

For $0\leq m\leq k$, define

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

Thus $\chi_0(k;K_n)=\omega(k;K_n)$ and $\chi_k(k;K_n)=\chi(k;K_n)$. Mixed decomposition conjecture. Let $m$ and $n\geq 1$ be non-negative integers. Then

$$\chi_m(k;K_n)\leq n+{k\choose 2}\quad\text{for all }k\geq m.$$

The source states that this bound for a given $m$ holds if and only if Plesník's conjecture holds for $k=m$, so it packages the chromatic decomposition conjecture into a family of mixed chromatic-clique bounds.