Covering a symmetric crossing supermodular function with hyperedges of prescribed size

Given a symmetric crossing supermodular function $p:2^V\to \mathbb{R}$ and positive integers $n_1,n_2,\dots,n_k$, does there exist a hypergraph H=(V,E) covering p and having exactly k hyperedges of sizes $n_1,n_2,\dots,n_k$?