Bounded twin-width three 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 of twin-width at most $3$. Twin-width conjecture. Does $\mathcal{G}$ have bounded tree-independence number? The paper notes that the analogous statement is known for twin-width $1$, while twin-width at most $4$ already permits unbounded tree-independence number; the case of twin-width at most $3$ is left open.