If I am not mistaken, the methods in the previous post (plus a conjecture about the ECH chain complex of perturbed boundaries of convex toric domains) can be used to show that if are real numbers with and , and if symplectically embeds into , and if , then . In other words, if you include one four-dimensional polydisk into another, such that the long sides are the same, and the short side of the domain is at least half the short side of the target, then this symplectic embedding is optimal.
The conjecture needed is that in the ECH chain complex of the (perturbed) boundary of a convex toric domain, a generator with only elliptic orbits represents a nontrivial homology class. (This would follow from a conjectural description of the differential in terms of rounding corners.) Without this conjecture, one can still prove a version of the above theorem in which the hypothesis is strengthened somewhat. (When you can still just assume .)
I’m working on writing this up cleanly.