The beta conjecture for maximal independent sets

Let $G$ be a graph on $n$ vertices, let $i(G)$ be the size of a smallest maximal independent set, and let $\tau(G)$ be the transversal number of the family of maximal independent sets. Define

$$\beta(G)=\frac{\tau(G)i(G)}{n}.$$

Beta conjecture. For all graphs $G$ on $n$ vertices,

$$\beta(G)=o(n).$$

This is presented as a reformulation of the Bollobás–Erdős–Tuza conjecture for $c$-hollow graphs. The paper notes that it holds trivially when $i(G)=o(n)$, while the general assertion remains open.