Coloring an interval graph with weights

I have an interval graph $G=(V,E)$ and a set of colors $C=\{c_1,c_2,...,c_m\}$, when a color $c_i$ is assigned to a vertex $v_j$, we have a score $u_{ij}\geq 0$. The objective is to find a coloring of $G$ with at most $m$ colors that maximizes the total score (sum of the scores of the vertices).

Do you know about any results in the literature that may help me to find the complexity of this problem?

The sum coloring problem can be reduced to this problem by having $n$ possible colors with weights $u_{ij} = n - i$. Sum coloring is NP-hard for interval graphs.

Source: https://en.wikipedia.org/wiki/Sum_coloring

StackExchange Computer Science Q#118760, answer score: 2