The quadratic Mader-number conjecture for bioriented cliques

Let $\accentset{\leftrightarrow}{K}_n$ denote the bioriented complete digraph on $n$ vertices, and let $\operatorname{mader}_{\vec{\chi}}(F)$ be the smallest integer $k\ge1$ such that every digraph of dichromatic number at least $k$ contains a subdivision of $F$. The quadratic Mader-number conjecture. There exists an absolute constant $c>0$ such that

$$\operatorname{mader}_{\vec{\chi}}(\accentset{\leftrightarrow}{K}_n)\le cn^2$$

for every positive integer $n$. The conjecture proposes a quadratic upper bound, improving substantially on the paper's general exponential upper bound; it remains open for general digraphs.