Erdős–Hajnal–Thomassen conjecture on high-chromatic subgraphs of prescribed girth and average degree

Let $k$ be a positive integer and let $g\ge3$ be an integer. For a graph, its chromatic number is denoted by $\chi(G)$, its average degree is the average of its vertex degrees, and its girth is the length of its shortest cycle. Erdős–Hajnal–Thomassen conjecture. For every positive integers $k$ and $g\ge3$, there exists a positive number $c(k,g)$ such that every graph with chromatic number at least $c(k,g)$ contains a subgraph of girth at least $g$ and with average degree at least $k$. The source presents this as a weakening of the Erdős–Hajnal and Thomassen conjectures and gives no resolution; it remains open.