Summary: The Gallai-Milgram theorem asserts that the vertex set of any digraph with stability number \(k\) can be partitioned into \(k\) directed paths. Hahn and Jackson conjectured that, for any positive integer \(k\), there exists a digraph with stability number \(k\) such that the subdigraph obtained by deleting any \(k-1\) directed paths still has stability number \(k\). They established the existence of such digraphs for \(k=1,2,3\). Here, we construct examples for arbitrarily large values of \(k\).