Linear topologies on Z are not Mackey topologies

Given a linear non-discrete topology λ on the integers, we show that there exists a strictly finer metrizable locally quasi-convex group topology τ on Z such that (Z,λ)∧=(Z,τ)∧ (algebraically). Applying this result to the p-adic topology on Z, we give a negative answer to the question of Dikranjan, asking whether this topology is Mackey.