The directed-clique conjecture for the minimum degree of digraphs

Let $D$ be a digraph, and let $\Delta_{\min}(D)$ denote its minimum degree parameter as defined in the paper. Let $\vec\chi(D)$ be its dichromatic number. A directed clique is a vertex set $X$ that can be partitioned into $(X_1,X_2)$ such that both $D[X_1]$ and $D[X_2]$ are complete digraphs and every possible arc from $X_1$ to $X_2$ is present. Let $\vec\omega(D)$ be the maximum size of a directed clique in $D$.

Directed-clique conjecture. There exists $\varepsilon>0$ such that every digraph $D$ satisfies

$$\vec\chi(D)\leq\left\lceil(1-\varepsilon)\Delta_{\min}(D)+\varepsilon\vec\omega(D)\right\rceil.$$

The conjecture is proposed after a related bound using the biclique number is shown to fail for $\Delta_{\min}$. It is presented as an open strengthening of the paper's partial result with $\widetilde{\Delta}$; no resolution is given.