The P=W identity for diffeomorphic cluster varieties and Lefschetz torus fibrations

Let $X$ be a 2-dimensional cluster variety, and let $h:Y\to\Delta$ be a Lefschetz $T^2$-fibration over the unit disk such that $Y$ is diffeomorphic to $X$. For a diffeomorphism $f:X\to Y$, write the induced pullback as $f^*:H^*(Y;\mathbb{Q})\to H^*(X;\mathbb{Q})$.

P=W conjecture. For any diffeomorphism $f:X\to Y$, the $P=W$ identity holds under $f^*$.

The conjecture asks whether the perverse filtration associated with the Lefschetz fibration is determined by the underlying smooth manifold of the cluster variety. The preceding theorem establishes the identity for elliptic fibrations in a specified deformation family, while the conjecture asserts it for every Lefschetz $T^2$-fibration satisfying the stated diffeomorphism condition.