Treewidth and the Product of Tree-Independence and Tree-Chromatic Numbers

For a graph $G$, let $\mathrm{tw}(G)$, $\mathrm{tree}\text{-}α(G)$, and $\mathrm{tree}\text{-}χ(G)$ denote its treewidth, tree-independence number, and tree-chromatic number, respectively. We prove that every nonempty graph $G$ satisfies $\mathrm{tw}(G)+1\leq\mathrm{tree}\text{-}α(G)\mathrm{tree}\te