The polyhedral symplectic cone conjecture for symplectic 4-manifolds

Let $W$ be a symplectic $4$-manifold. A polyhedral symplectic cone is a (possibly non-compact) polytope $C\subset H^2(W;\mathbb{R})$ whose dual polytope lies in $H_2(W;\mathbb{R})$.

Polyhedral symplectic cone conjecture. There exists a polyhedral symplectic cone $C$ with the following properties:

1. The dual polytope in $H_2(W;\mathbb{R})$ is compact, symmetric, convex and integral.
2. There exist open top-dimensional faces $F_1,\dots,F_s$ of $C$ such that the symplectic cone coincides with all non-degenerate elements in the cone on $F_1,\dots,F_s$.

This conjecture proposes that the symplectic cone of every symplectic $4$-manifold has a structure analogous to the fibered cone of a $3$-manifold. The source does not state a resolution, so the conjecture remains open.