3-colorability conjecture for string graphs of sufficiently large odd girth

Let $k$ be an integer, and consider the class of string graphs whose odd girth is at least $k$. String-graph coloring conjecture. There is an integer $k$ such that the class of string graphs of odd girth at least $k$ is 3-chromatic. While string graphs of large odd girth can contain arbitrarily large bipartite complete graphs and therefore are not $d$-degenerate for any fixed $d$, the source gives no known proof or disproof of this 3-colorability assertion.