  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.
