Extremal sum-color cost conjecture for k-trees

A $k$-tree is a graph obtained from $K_k$ by iteratively adding a vertex whose neighborhood is a $k$-clique in the existing graph. The join $G\vee H$ is obtained from the disjoint union $G+H$ by making every vertex in $G$ adjacent to every vertex in $H$. The $k$th power $P_n^k$ of the path $P_n$ has the same vertex set as $P_n$, with two vertices adjacent exactly when their distance in $P_n$ is at most $k$. For $k\in\mathbb{N}$, let $T$ be a $k$-tree with $n$ vertices, and let $\operatorname{\mathaccent"017{s}}(G)$ denote the sum-color cost of a graph $G$. Extremal sum-color cost conjecture for $k$-trees.

$$\operatorname{\mathaccent"017{s}}(K_k\vee \overline{K}_{n-k})\leq\operatorname{\mathaccent"017{s}}(T)\leq\operatorname{\mathaccent"017{s}}(P_n^k).$$

This extends the proved extremal result for trees, with the split graph $K_k\vee\overline{K}_{n-k}$ conjectured to minimize and the $k$th power of the path conjectured to maximize the sum-color cost among $n$-vertex $k$-trees. The claim is presented as a conjecture, and no resolution is given in the source.