The treewidth-linear basis number conjecture

Let $G$ be a graph, let $k$ be a nonnegative integer, and let $\operatorname{bn}(G)$ denote the basis number of $G$. Treewidth-linear basis number conjecture. Every graph $G$ of treewidth at most $k$ has

$$\operatorname{bn}(G)\in O(k).$$

This conjecture would improve the known bounds for graphs of bounded treewidth. The paper states that the arguments presented were tried unsuccessfully so far, and does not report a resolution.