Bounded sim-width versus tree-independence number for star-free graph classes

Let $d\geq 2$, and let $\mathcal{G}$ be a $K_{1,d}$-free graph class. Sim-width conjecture. If $\mathcal{G}$ has bounded sim-width, then $\mathcal{G}$ has bounded tree-independence number. The question is presented as a weaker version of the corresponding comparison problem for $K_{t,t}$-free graph classes; it remains open in the paper.