Nagai's conjecture on monodromy nilpotency for hyper-Kähler varieties

Let $\Delta$ be the open unit disk, let $\pi\colon\mathcal{X}\to\Delta$ be a degeneration of hyper-Kähler varieties of dimension $2n$ over $\Delta^*=\Delta\setminus\{0\}$, and let $X=\pi^{-1}(t)$ be a smooth fiber for $t\in\Delta^*$. For each $i$, let $N_i$ be the (log-)monodromy operator on the Betti cohomology $H^i(X,\mathbb{Q})$. For a nilpotent linear operator $N\in\operatorname{End}(V)$, define its nilpotency index by

$$\nu(N)\coloneqq\min\{m\in\mathbb{Z}_{\geq 0}\mid N^{m+1}=0\}.$$

Nagai's conjecture. The monodromy operators satisfy

$$\nu(N_{2i})=i\nu(N_2)$$

for every $0\leq i\leq n$. Nagai's conjecture predicts a precise relation among monodromy nilpotency indices in the even-degree cohomology of degenerating hyper-Kähler varieties; its arithmetic analogues are formulated for $\ell$-adic and $p$-adic monodromy operators. The source gives no evidence of a resolution.