*Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into a Zoll two-sphere?*

I'm also interested in the same question on the $n$-disc. A related question is whether one can distinguish a Zoll metric locally (or at least say: this metric cannot be Zoll because in the neighborhood of this point it does not behave in such and such way). I'm guessing the answer to this last question is **no** and this prompted the first question.