Divisibility results for zero-cycles

Let X be a product of smooth projective curves over a finite unramified extension k of Q(p). Suppose that the Albanese variety of X has good reduction and that X has a k-rational point. We propose the following conjecture. The kernel of the Albanese map CH0(X)(0) -> Alb(X)(k) is p-divisible. When p is an odd prime, we prove this conjecture for a large family of products of elliptic curves and certain principal homogeneous spaces of abelian varieties. Using this, we provide some evidence for a local-to-global conjecture for zero-cycles of Colliot-Thelene and Sansuc (Duke Math J 48(2):421-447, 1981), and Kato and Saito (Contemporary Mathematics, vol. 55:255-331, 1986).