Minimal truncations of supersingular p-divisible groups

Let k be an algebraically closed field of characteristic p>0. Let H be a supersingular p-divisible group over k of height 2d. We show that H is uniquely determined up to isomorphism by its truncation of level d (i.e., by H[p^d]). This proves Traverso's truncation conjecture for supersingular p-divisible groups. If H has a principal quasi-polarization \lambda, we show that (H,\lambda) is also uniquely determined up to isomorphism by its principally quasi-polarized truncated Barsotti--Tate group of level d (i.e., by (H[p^d],\lambda[p^d])).