Multiplicative decomposition and Chern-class conjecture for Lagrangian fibrations

Let $f:M\to B$ be a Lagrangian fibration: $M$ and $B$ are nonsingular, $f$ is proper and flat, and a general fiber is an abelian variety of half the dimension of $M$, with a holomorphic symplectic form on $M$ restricting to zero on the general fiber.

Lagrangian-fibration conjecture. The morphism $f$ admits a multiplicative decomposition theorem

$$Rf_*\mathbb{Q}_M\simeq\bigoplus_i\mathcal{H}_{(i)},\qquad \mathcal{H}_{(i)}:={}^{\mathfrak p}H^{i+\dim B}(Rf_*\mathbb{Q}_M)[-i-\dim B]\in D^b_c(B),$$

and, for the induced cohomological decomposition

$$H^*(M,\mathbb{Q})=\bigoplus_i H^*_{(i)}(M,\mathbb{Q}),$$

the Chern classes satisfy

$$c_i(T_M)\in H^{2i}_{(i)}(M,\mathbb{Q}).$$

The claim packages two expected properties of Lagrangian fibrations: multiplicativity of the decomposition and compatibility of the tangent-bundle Chern classes with its grading. The supplied text does not state whether either assertion has been proved in general.