Skew-supermodular colouring with two class sizes

Let [math]p_1[/math] and [math]p_2[/math] be integer skew-supermodular set functions on ground set S such that [math]\max\{p_1(X),p_2(X)\}\leq \min\{|X|,k\}[/math] for every [math]X \subseteq S[/math], and let [math]m_1,m_2,n_1,n_2[/math] be positive integers such that [math]m_1 n_1+m_2 n_2=|S|[/math]. Can we decide in polynomial time if there is a partition [math]{\mathcal P}[/math] of S with [math]m_1[/math] classes of size [math]n_1[/math] and [math]m_2[/math] classes of size [math]n_2[/math], such that [math]|\{Y \in {\mathcal P}: Y \cap X \neq \emptyset\}| \geq \max\{p_1(X), p_2(X)\}[/math] for every [math]X \subseteq S[/math]?

Remarks

The problem is open even in the special case if the two functions [math]p_1[/math] and [math]p_2[/math] are intersecting supermodular. If [math]|n_1-n_2| \leq 1[/math], then the partition always exists (and can be found in polynomial time) by the skew-supermodular variant of Schrijver's Supermodular colouring theorem. Another known special case is the result of Folkman and Fulkerson that characterizes bipartite graphs that can be partitioned into [math]m_1[/math] matchings of size [math]n_1[/math] and [math]m_2[/math] matchings of size [math]n_2[/math]. For three distinct class sizes, the problem becomes NP-complete , see Skew-supermodular colouring with prescribed class-sizes.