Plectic comparison conjecture for partial Frobenius and filtrations

Let $f$ be the Hilbert modular form and let $D_p(f)$ be its $p$-adic de Rham cohomology with partial Frobenius operators $\varphi_1,\dots,\varphi_d$ and partial filtrations $\operatorname{Fil}_i^\bullet$. Choose a global isomorphism

$$\psi: V_p(f)\cong (\bigotimes\text{-Ind})(V_p^{\mathrm{std}}(f))$$

and let

$$\psi_p:D_p(f)\cong\bigotimes_{i=1}^dD_{\mathfrak{p}_i}(f)$$

be the induced isomorphism of filtered $\varphi$-modules, where $D_{\mathfrak{p}_i}(f)$ is the filtered $\varphi$-module of the standard representation at $\mathfrak{p}_i$. Plectic comparison conjecture. For some choice of global isomorphism $\psi$ as above, and each $i=1,\dots,d$, the isomorphism $\psi_p$ intertwines the partial Frobenius $\varphi_i$ on $D_p(f)$ with the operator

$$1\otimes\dots\otimes1\otimes\varphi\otimes1\otimes\dots\otimes1$$

with $\varphi$ in the $i$-th component, on $\bigotimes_iD_{\mathfrak{p}_i}(f)$; similarly, it intertwines the $i$-th partial filtration $\operatorname{Fil}_i^\bullet$ on $D_p(f)$ with

$$D_{\mathfrak{p}_1}(f)\otimes\dots\otimes D_{\mathfrak{p}_{i-1}}(f)\otimes(\operatorname{Fil}^\bullet D_{\mathfrak{p}_i}(f))\otimes D_{\mathfrak{p}_{i+1}}(f)\otimes\dots\otimes D_{\mathfrak{p}_d}(f).$$

The conjecture would identify the plectic structures on the cohomology of the Hilbert modular variety with the factorwise structures arising from tensor induction. The preceding partial Eichler--Shimura relation gives compatible polynomial relations for the partial Frobenii, but the simultaneous comparison with the tensor factors and partial filtrations remains conjectural.