Geometric P=W conjecture for Lagrangian tori

Let \mathcal{X}\xymatrix@1@=15pt{\ar[r]&}\Delta be a projective minimal divisorial log terminal type III degeneration of hyperkähler manifolds, let $\mathcal{X}_t$ be a smooth fiber, and let $\mathbb{T}\subset\mathcal{X}_t$ be a profound torus. Let f\colon X\xymatrix@1@=15pt{\ar[r]&} B be a Lagrangian fibration with general fiber $T$. Geometric P=W conjecture. For every such fibration, there exists a degeneration \pi\colon\mathcal{X}\xymatrix@1@=15pt{\ar[r]&}\Delta with $\mathcal{X}_t$ deformation equivalent to $X$ for all $t\in\Delta^*$ such that $T$ is isotopic to a profound torus $\mathbb{T}$.

$$T\simeq_{\mathrm{isotopic}}\mathbb{T}.$$

This conjecture proposes a geometric explanation for the equality between perverse and weight Hodge numbers. The required degeneration and isotopy are not established in general.