The compact hyperkähler P=W conjecture

Let $M$ be a compact hyperkähler manifold admitting a holomorphic Lagrangian torus fibration $\ell:M\to B$, and let $\beta$ be the pullback of an ample divisor on $B$. Let $M_1$ be a deformation of $M$ near the type III boundary point determined by $\beta$, and let $W_\bullet H^k(M_1;\mathbb{Q})$ be the monodromy weight filtration associated with the corresponding limit mixed Hodge structure. The compact hyperkähler P=W conjecture. There is a diffeomorphism between $M$ and $M_1$ such that

$$P_iH^k(M;\mathbb{Q})=W_{2i}H^k(M_1;\mathbb{Q})=W_{2i+1}H^k(M_1;\mathbb{Q})$$

for all $i$ and $k$. This is proposed as a compact analogue of the P=W conjecture, relating the perverse Leray filtration of a Lagrangian fibration to the monodromy weight filtration of a degeneration. The source gives no resolution.