The $p$-adic analogue of Nagai's conjecture

Fix a prime $p$, let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field, and let $X$ be a hyper-Kähler variety over $K$ of dimension $2n$. The potentially semistable representation $H^i_{\mathrm{\acute{e}t}}(X_{\overline K},\mathbb{Q}_p)$ has a Fontaine monodromy operator $N_{p,i}$ on

$$\operatorname{D_{pst}}(H^i_{\mathrm{\acute{e}t}}(X_{\overline K},\mathbb{Q}_p)).$$

Here $\operatorname{D_{pst}}$ is Fontaine's functor for potentially semistable representations. The $p$-adic analogue of Nagai's conjecture. The monodromy operators satisfy

$$\nu(N_{p,2i})=i\nu(N_{p,2})$$

for every $0\leq i\leq n$. This conjecture is the arithmetic $p$-adic counterpart of Nagai's complex-geometric conjecture. The source gives no evidence of a resolution.