Covering a symmetric crossing supermodular function with hyperedges of prescribed size

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

Remarks

If the sizes are all equal then the problem was solved by Tamás Király .