The symplectic cone polytope conjecture

Let $W$ be a symplectic $4$-manifold. The symplectic cone is the set of all classes in $H^2(W;\mathbb{R})$ represented by symplectic forms. Let $C\subset H^2(W;\mathbb{R})$ be a possibly non-compact polytope, and let $F_1,\dots,F_s$ be open top-dimensional faces of $C$.

Symplectic cone polytope conjecture. There exists such a polytope $C$ satisfying:

1. Its dual polytope in $H_2(W;\mathbb{R})$ is compact, symmetric, convex and integral.
2. The symplectic cone is precisely the set of all non-degenerate elements in the cone on $F_1,\dots,F_s$.

The conjecture is motivated by the preceding determination of the symplectic cone for a significant class of $4$-manifolds and by its expected analogy with the fibered cone of a $3$-manifold. The source does not state a resolution.