The cluster-variety P=W and PI=WI conjectures

Let $X$ be a cluster variety over ${\mathbb{C}}$, and write $X(M)$ for the corresponding character variety. A real Lagrangian fibration is a map

$$h:X\to {\mathbb{R}}^{\dim_{\mathbb{C}} X}.$$

P=W and PI=WI conjectures. There exists a real Lagrangian fibration $h$ such that

$$P_kH^*(X(M),{\mathbb{Q}})=W_{2k}H^*(X(M),{\mathbb{Q}})=W_{2k+1}H^*(X(M),{\mathbb{Q}}),\qquad k\ge 0.$$

Moreover, the analogous identity remains true with cohomology replaced by intersection cohomology $IH^*(X,{\mathbb{Q}})$.

These conjectures extend the $P=W$ phenomenon from character varieties to cluster varieties and propose a corresponding statement for intersection cohomology. The supplied material gives no evidence that either assertion has been resolved.