Positive vector measures with given marginals

Suppose E is an ordered locally convex space, X ₁ and X ₂ Hausdorff completely regular spaces and Q a uniformly bounded, convex and closed subset of M _t ⁺ (X ₁ × X ₂, E). For i = 1, 2, let μ _i   M _t ⁺ (X _i , E). Then, under some topological and order conditions on E, necessary and sufficient conditions are established for the existence of an element in Q, having marginals μ ₁ and μ ₂.