Treewidth-bounded Eulerian digraph immersion conjecture

Let $k\in\mathbb{N}$, and consider the class of Eulerian digraphs whose underlying undirected graphs have treewidth at most $k$. Let immersion mean the immersion relation without requiring the immersions to be strong.

Treewidth-bounded immersion conjecture. The class of Eulerian digraphs of treewidth at most $k$ is well-quasi-ordered by immersion.

The conjecture is proposed as a first step toward extending the positive bounded-degree strong-immersion result to classes allowing unbounded degree, complementing the strong-immersion antichain for planar Eulerian digraphs of treewidth at most $2$. The source gives no resolution.