The biclique corollary for the minimum degree of digraphs

Let $D$ be a digraph. Let $\Delta_{\min}(D)$ denote the paper's minimum degree parameter, let $\vec\chi(D)$ be its dichromatic number, and let \overset{\text{\tiny\bm\leftrightarrow}}{\omega}(D) be its biclique number, the maximum order of a set inducing a complete digraph.

Biclique corollary conjecture. There exists $\varepsilon>0$ such that every digraph $D$ satisfies

$$\vec\chi(D)\leq\left\lceil(1-\varepsilon)\Delta_{\min}(D)+2\varepsilon\overset{\text{\tiny$\bm\leftrightarrow$}}{\omega}(D)\right\rceil.$$

The paper says this weaker conjecture follows from the directed-clique conjecture and would show that the condition $\varepsilon'<2\varepsilon$ in the preceding counterexample discussion is necessary. It remains open in the supplied text.