How much of a ball $B_R^{2n}$ can be symplectically embedded in the cylinder $B_r^2\times\mathbb R^{2n-2}$? Gromov's nonsqueezing theorem famously says that there does not exist any symplectic embedding $B^{2n}_R\hookrightarrow Z^{2n}_r:=B^2_r\times\mathbb R^{2n-2}$ when $R>r$, but can we make this non-existence quantitative? I suspect we should have a result like follows:
Conjecture: For any symplectomorphism $f:\mathbb R^{2n}\to\mathbb R^{2n}$ and any $R\ge r$, we have a uniform bound $$\operatorname{Vol}\left(f(B_R^{2n})\cap Z_r^{2n}\right)\le C R^{2n-2}r^2.$$
If a result holds, I would expect it to be fairly difficult to prove – it would seem to require a very different proof than Gromov nonsqueezing, which works by integrating along a suitable $J$-holomorphic $S^2$ and bounding its energy from below by using blowup at the embedded ball. My main question is then, has a result like this been proved or analyzed somewhere in the literature?
Edit: This statement can be restated as follows.
Conjecture: Any set $A\subset B_R^{2n}$ of symplectic capacity $c(A)\le1$ has volume $\operatorname{Vol}(A)\le C_n R^{2n-2}$.
 A: I had thought about a quantitative version of this for a while, so let me say what I know.
The first point to make is that the conjecture as stated is incorrect. In fact, $\operatorname{Vol}(f(B_R) \cap Z_r)$ can be arbitrarily close to $\operatorname{Vol}(B_R)$. This is due to Anatole Katok in his paper Ergodic Perturbations of Degenerate Integrable Hamiltonian Systems from 1973: see the Basic Lemma of Section 3 (many thanks to Umut Varolgunes for pointing this out to me). The idea is something like this: break $B_R$ into a bunch of small cubes separated by very thin walls, so that the volume of those walls is as close to 0 as you want. Now you can move the small cubes around by translation via Hamiltonians (which you cut off inside the walls to produce global Hamiltonian diffeomorphisms). So just translate all of the cubes into the cylinder, and you're done.
This might sound discouraging, but in fact, there is a very precise quantitative question to ask, which is how arbitrarily close to $\operatorname{Vol}(B_R)$ can we be if $f$ has a bounded Lipschitz constant. (I learned this problem from Larry Guth, but I suspect it was in the ether beforehand.) To be precise, I'm going to fix $R>r$, and define $$c_n(L) = \inf_{\|f\|_{Lip} \leq L}\{\operatorname{Vol}(B^{2n}_R\setminus f^{-1}(Z^{2n}_r))\}$$ (where implicitly in the infimum we only take symplectic embeddings). Then I can prove that $$O(L^{-2n+1}) \leq c_n(L) \leq O(L^{-1}).$$ A lower bound of $O(L^{-2n})$ is a nice exercise (from which it's not hard to gain an extra factor of $L$), and the upper bound is a careful version of Katok's construction. I suspect one can use $J$-holomorphic curves to improve the lower bound, but I (and others I discussed this problem with) found it to be quite difficult. I'm happy to discuss these ideas with anyone interested - I think it'd be a lovely result if one could understand the asymptotics of $c_n(L)$ much better, and as far as I know, such quantitative non-squeezing doesn't really appear in the literature anywhere.
