Directed disjoint paths remains W[1]-hard on acyclic digraphs without large grid minors

In the Vertex Disjoint Paths with Congestion problem, the input consists of a digraph D, an integer c and k pairs of vertices (sᵢ, tᵢ), and the task is to find a set of paths connecting each sᵢ to its corresponding tᵢ, whereas each vertex of D appears in at most c many paths. The case where $c = 1$ is known to be NP-complete even if $k = 2$ [Fortune, Hopcroft and Wyllie, 1980] on general digraphs and is W[1]-hard with respect to k (excluding the possibility of an f(k)nO(1)-time algorithm under standard assumptions) on acyclic digraphs [Slivkins, 2010]. The proof of [Slivkins, 2010] can also be adapted to show W[1]-hardness with respect to k for every congestion c ≥ 1. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion c ≥ 1 even if: - the input digraph D is acyclic, - D does not contain an acyclic $(5, 5)$-grid as a butterfly minor, - D does not contain an acyclic tournament on 9 vertices as a butterfly minor, and - D has ear-anonymity at most 5. Further, we also show that the edge-congestion variant of the problem remains W[1]-hard for every congestion c ≥ 1 even if: - the input digraph D is acyclic, - D has maximum undirected degree 3, - D does not contain an acyclic $(7, 7)$-wall as a weak immersion and - D has ear-anonymity at most 5.