BNY property conjecture for digraphs

Let $D$ be a digraph. A $k$-path subdigraph of $D$ is a collection of $k$ vertex-disjoint paths, and let $\lambda_k(D)$ denote the maximum order of a $k$-path subdigraph; in particular, $\lambda(D)=\lambda_1(D)$. A digraph $D$ satisfies the BNY property if, for every integer $q$ with $1\leq q\leq\lambda(D)-1$, there is a partition $(A,B)$ of $D$ such that $\lambda(A)\leq q$ and, for all $k\in\{1,2,\dots,|V(B)|\}$, $\lambda_k(B)\leq\lambda_k(D)-q$. BNY property conjecture. Property BNY holds for every digraph. The BNY property is a strengthening of the Path Partition Conjecture, and the paper proves it for wide families of acyclic and semicomplete compositions; the assertion for arbitrary digraphs remains open.