Network Structures

Consider the set-up for the bipartite graph auction, with an equal number of buyers and sellers, and with each buyer having a valuation for the object being sold by each seller. Suppose that we have an instance of this problem in which there is a particular seller i who is the favorite: every buyer j has a strictly higher valuation for seller is object than for the object being sold by any other seller k. (In notation, we have vij > vkj for all choices of j and k 6= i.) Consider a set of market-clearing prices in this situation. Must it be the case that the price charged by seller i is strictly higher as the price charged by any other seller?

(HINT: This appears to be the case. Suppose that seller i has an object that is most valuable to all. Suppose there are market clearing prices such that buyer j buys seller ks object paying at least as much as the price of the is object. Is this an edge in the graph of the preferred buyers?)