Summary: In this paper, we seek to design a Groves mechanism for assigning $p$ heterogeneous objects among $n$ competing agents ($n > p)$ with unit demand, satisfying weak budget balance, individual rationality, and minimizing the budget imbalance. This calls for designing an appropriate rebate function. When the objects are identical, this problem has been solved by Moulin [1] and Guo and Conitzer [2]. However, it remains an open problem to design such a rebate function when the objects are heterogeneous. We propose a mechanism, HETERO and conjecture that HETERO is individually rational and weakly budget balanced. We provide empirical evidence for our conjecture through experimental simulations.