Dallard et al.'s induced-grid characterization conjecture for star-free graph classes

Let $d$ be a positive integer and let $\mathcal{G}$ be a hereditary $K_{1,d}$-free graph class. Dallard et al.'s conjecture. The class $\mathcal{G}$ has bounded tree-independence number if and only if there exists a positive integer $k$ such that $\mathcal{G}$ excludes the $(k\times k)$-grid as an induced minor. This would generalize the induced Grid-Minor Theorem for graphs of bounded maximum degree; the paper presents it as a conjecture and does not report a resolution.