Polynomial bound for tree-independence number in star-free induced-grid-minor-free graphs

Let $d\geq 2$ be an integer, let $G$ be a $K_{1,d}$-free graph, and let $f$ be a polynomial. Polynomial-bound conjecture. There is a polynomial $f$ such that, whenever $G$ does not contain a $d\times d$ grid as an induced minor, its tree-independence number satisfies

$$\alpha\text{-}\mathsf{tw}(G)\leq f(d).$$

The question strengthens the induced-grid characterization by asking for a polynomial bound in $d$; the paper states it as an open problem and gives no resolution.