On minimal arbitrarily partitionable graphs

A graph G=(V,E) of order n is called arbitrarily partitionable, or AP for short, if given any sequence of positive integers n"1,...,n"k summing up to n, we can always partition V into subsets V"1,...,V"k of sizes n"1,...,n"k, resp., inducing connected subgraphs in G. If additionally G is minimal with respect to this property, i.e. it contains no AP spanning subgraph, we call it a minimal AP-graph. It has been conjectured that such graphs are sparse, i.e., there exists an absolute constant C such that |E|==1+130 (if such C exists).