Consider the open set $M \subset \mathbb{C}^{2}$ given by the union of the unit ball $|z_1|^2 + |z_{2}|^2 < 1$ (the coconut) and the cylinder $|z_1| < \epsilon$, $0 < \epsilon < \! \!< 1$, (the straw, which in this case pierces the coconut through and through, but this is not important).

Fix a number $r$ strictly between $\epsilon$ and $1$.

Does there exist a strictly positive number $c$ so that the volume of the intersection of the unit ball with any symplectic image of the ball of radius $r$ lying wholly inside $M$ is greater than $c$? If so, is there a reasonable estimate for $c$ as a function of $r$ and $\epsilon$?

By Gromov's non-squezing theorem we know we cannot symplecticaly move the whole ball of radius $r$ up the straw, but it is not clear to me how much of its volume can we sip out of the coconut.

This problem is related to the comments I got on this question.