A conjecture on convex symmetric curve. How to prove it?

[KaiJin]

Suppose Q is a central symmetric convex planar curve. We define f(Q) to be max{Area(P)/Area(Q) | "P is a parallelogram inside Q."}

And we want to find f_min= min{f(Q)}.

It's trivial that f_min > 1/2. And with some exploration I can prove f_min> 4/(4+pi)=0.56 .
On the other hand, for any ellipse Q, f(Q)=2/pai. so f_min <=2/pai=0.6366 .

By notice that from any point on a ellipse we can draw a parrallelogram inside it whose area is global optimal (All equal to 2*a*b), it is natual to guess that f_min=2/pai.

Now I am quite sure of the correctness of this conjecture. But I don't know how to prove it!
I think it's very interesting. Could somebody help me?