Linial's path-partition conjecture for digraphs

Let $D$ be a digraph and let $k$ be a positive integer. The $k$-norm of a path partition $\mathcal{P}$ of $D$ is $|\mathcal{P}|_k=\sum_{P\in\mathcal{P}}\min\{|V(P)|,k\}$. A path partition is $k$-minimum when its $k$-norm is minimum, and this minimum is denoted by $\pi_k(D)$. A partial $k$-coloring is a collection of $k$ vertex-disjoint stable sets; let $\alpha_k(D)$ be the maximum number of vertices covered by such a coloring.

Linial's conjecture. For every digraph $D$ and every positive integer $k$,

$$\pi_k(D)\leq\alpha_k(D).$$

This extends the Gallai–Milgram inequality $\alpha_1(D)\geq\pi_1(D)$ and is one of Linial's two proposed generalizations involving $k$-optimal path partitions and colorings. The source states that the conjecture remains open.