Thomassen's conjecture on dense bipartite subgraphs of large girth

Let $t,k\in\mathbb{N}$. A bipartite graph has average degree at least $f(t,k)$ if its average degree is at least the value of a function $f:\mathbb{N}^2\to\mathbb{N}$.

Thomassen's conjecture. There is a function $f:\mathbb{N}^2\rightarrow\mathbb{N}$ such that, for all $t,k\in\mathbb{N}$, every bipartite graph of average degree at least $f(t,k)$ has a subgraph of average degree at least $t$ and girth at least $k$.

This is a conjecture about finding arbitrarily dense subgraphs of arbitrarily large prescribed girth in bipartite graphs. The paper notes that the case of girth $6$ is currently the best known.