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 $\partial X$ admits a Reeb chord with distinct endpoints. A higher dimensional analog is also discussed.