Kanté and Kwon's tree vertex-minor conjecture for linear rank-width

Let $T$ be a tree. A graph is $T$-vertex-minor-free if it does not contain $T$ as a vertex-minor, and linear rank-width is the graph width parameter used in the claim.

Kanté and Kwon's conjecture. For every tree $T$, the class of $T$-vertex-minor-free graphs has bounded linear rank-width.

The conjecture is known for graph classes whose prime graphs under split decompositions have bounded linear rank-width, including distance-hereditary graphs, and for every tree $T$ that is a path. The full assertion remains open.