 Arbitrarily large symplectic balls in a cotangent cylinder

The symplectic manifold $T^2\times {\mathbb{R}}^2$ contains symplectic balls of arbitrarily large radius. A steep linear Lagrangian graph in ${\mathbb{R}}^4=T^{\ast }{\mathbb{R}}^2$ has a thin neighborhood that injects under the quotient ${\mathbb{R}}^2\to T^2$, while a symplectic base-fiber dilation fits a large round ball into that long thin neighborhood.