Biedl's logarithmic basis number conjecture

Let $G$ be an $n$-vertex graph, and let $\operatorname{bn}(G)$ denote the basis number of $G$. Biedl's logarithmic basis number conjecture. Every $n$-vertex graph $G$ has

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

The conjecture proposes a logarithmic upper bound on the basis number for arbitrary graphs. The source presents it as an unresolved conjecture and attributes it to Biedl; it suggests that insights from the paper's proof may help establish it.