Torsion zero-cycles on a product of canonical lifts of elliptic curves

Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We write T (X) and T (Y) for the kernels of the Albanese maps of X and Y, respectively. Then, F(X) = T(X)/T(X)_div is conjectured to be finite, where T(X)_div is the maximal divisible subgroup of T(X). Furthermore, F(X) is conjectured to be isomorphic to T(Y) modulo p-primary torsion. We show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y.