Esperet–Kang–Thomassé conjecture on dense induced bipartite subgraphs

Let $d$ be a positive real number, and let $G$ be a triangle-free graph with minimum degree at least $d$. An induced bipartite subgraph of $G$ is a bipartite subgraph induced by a subset of the vertices of $G$, and its average degree is the average of its vertex degrees. Esperet–Kang–Thomassé conjecture. The graph $G$ contains an induced bipartite subgraph with average degree

$$\Omega(\ln d).$$

This conjecture strengthens the known bound of order $\ln d/\ln\ln d$ proved for every triangle-free graph of minimum degree $d$. It asks whether the logarithmic factor can be achieved without the additional $\ln\ln d$ loss.