The wall-subdivision formulation of the induced-grid conjecture

Let $d$ be a positive integer and let $\mathcal{C}$ be a hereditary graph class excluding $K_{1,d}$. A subdivision of the elementary $k$-wall is obtained by subdividing edges of the elementary $k$-wall, and its line graph is formed by taking edges as vertices. The wall-subdivision formulation. The class $\mathcal{C}$ has bounded tree-independence number if and only if there exists a positive integer $k$ such that $\mathcal{C}$ excludes all subdivisions of the elementary $k$-wall and their line graphs. This is stated as an equivalent induced-subgraph formulation of the induced-grid conjecture and remains open with that conjecture.