Strong Arnold chord conjecture via normalized capacities

We show that every dynamically convex toric domain in ${\mathbb{R}}^4$ admits a $C^1$-neighborhood $\mathcal{U}$ within the space of star-shaped domains of ${\mathbb{R}}^4$ with the following property: for any $X\in \mathcal{U}$, every Legendrian knot in $\mathrm{\partial }X$ admits a Reeb chord with distinct endpoints. A higher dimensional analog is also discussed.