The bioriented-cycle subdivision conjecture

Let $D$ be a digraph, let $\vec{\chi}(D)$ denote its dichromatic number, and let $\accentset{\leftrightarrow}{C}_\ell$ denote the bioriented cycle of length $\ell$. Bioriented-cycle subdivision conjecture. If $D$ has $\vec{\chi}(D)\ge3$, then there exists an integer $\ell\ge3$ such that $D$ contains a subdivision of $\accentset{\leftrightarrow}{C}_\ell$. Equivalently, the maximum dichromatic number of a digraph containing no subdivision of any bioriented cycle should be $2$. This remains an open question about forcing bioriented-cycle subdivisions in digraphs.