Mader's conjecture on openly disjoint cycles in regular digraphs

For a digraph $D$, let $c(D)$ be the largest integer $k$ for which there are directed cycles $C_1,\ldots,C_k$ through a common vertex $v$ such that $C_1-v,\ldots,C_k-v$ are pairwise vertex-disjoint. For $r\in\mathbb{N}$, let $c_r$ be the minimum of $c(D)$ over all $r$-regular digraphs. Mader's conjecture. For every $k\in\mathbb{N}$ there exists some $r_0\in\mathbb{N}$ such that $c_r\ge k$ for every $r\ge r_0$; equivalently,

$$\lim_{r\rightarrow\infty}c_r=\infty.$$

The paper proves this conjecture in the stronger form $c_r\ge \lceil\frac{3}{22}r\rceil$ for every $r\in\mathbb{N}$, so the conjecture is solved.