Matsushita's maximal-or-minimal variation conjecture for Lagrangian fibrations

Let $\pi:X\rightarrow\mathbb{P}^n$ be a Lagrangian fibration, and let $t\in\mathbb{P}^n$ be a generic point. After choosing a local section near $t$, let $\phi$ be the resulting local classifying map to $\mathcal{A}_{d_1,\ldots,d_n}$. The fibration has maximal variation if the image of $\phi$ is $n$-dimensional, equivalently if $(d\phi)_t$ has rank $n$, and it has minimal variation if $\phi$ is constant and $d\phi$ vanishes.

Matsushita's conjecture. The fibration has either maximal variation or minimal variation.

For elliptic K3 surfaces, failure of maximal variation forces the classifying map to be constant, and examples are locally isotrivial. The conjecture proposes the same dichotomy in higher dimensions; the source gives no resolution.