Double implementation in a market for indivisible goods with a price constraint

I consider the problem of assigning indivisible objects to agents, in which each agent pays a price for his object and all prices sum to a given constant. The objective is to select an assignment-price pair that is envy-free with respect to the agents' true preferences. I propose a simple mechanism whereby agents announce valuations for all objects and an envy-free allocation is selected with respect to these announced preferences. I prove that the proposed mechanism implements both in Nash and strong Nash equilibrium the set of true envy-free allocations.