The induced Grid Theorem for $K_{1,d}$-free graphs

Let $k,d$ be positive integers, and let $G$ be a graph. A graph is $K_{1,d}$-free if it has no induced subgraph isomorphic to $K_{1,d}$. The tree-independence number $\alpha$-$\mathsf{tw}(G)$ is the minimum, over all tree-decompositions of $G$, of the maximum independence number of a bag.

The induced Grid Theorem. There exists a function $f:\mathbb{N}^2\rightarrow\mathbb{N}$ such that every $K_{1,d}$-free graph $G$ with $\alpha$-$\mathsf{tw}(G)\geq f(k,d)$ contains the $(k\times k)$-grid as an induced minor.

This is the conjectured induced analogue of the Grid Theorem for graphs of bounded induced degree. The paper discusses it as a target for the results developed here; its resolution remains open.