There are three approaches,

• Daping's canonical skeleton using $\omega = Im(\Omega)$.
• Constructing Kahler potential by extending (after modification) standard KP on $(\C^*)^n$.
• Semi-Tropicalize the cluster variety, put the main torus on some flat region. Let trickle down from

They each need to solve some technical problems.

There is a canonical subspace, that is easy to specify, and easy to check that it is a skeleton. The trouble is, how to show there is a Liouville flow that realize this as a Liouville skeleton.

• (OK) One can engineer the flow near the skeleton, that is a local gluing / plumbing problem
• (Hard) Find a Morse function whose critical manifold is as the skeleton.