Venkatesh's $p$-adic motivic rationality conjecture

Let $\pi$ be as in the $p$-adic cohomological setting, let $\text{\dn{v}}_{\pi,p}$ be the $p$-adic motivic cohomology space, and let $\text{\dn{v}}^*_{\pi,E(\pi_f)}\subset\text{\dn{v}}^*_{\pi,p}\otimes\mathbf Q_p$ be the subspace whose pairings with motivic cohomology lie in $E(\pi_f)$. Let $H^*(S(G,X),\widetilde V)_\pi$ be the $\pi_f$-isotypic subspace and $K_0$ the specified level subgroup. Venkatesh's $p$-adic rationality conjecture. Under the action of $\bigwedge^\bullet\text{\dn{v}}^*_{\pi,p}$ on

$$[H^*(S(G,X),\widetilde V)_\pi]^{K_0}\otimes\mathbf Q_p\subset H^*(S_{K_0},\Lambda_V)_\mathfrak m\otimes\mathbf Q_p,$$

induced by the motivic action, the action of $\text{\dn{v}}^*_{\pi,E(\pi_f)}$ preserves the $E(\pi_f)$-rational structure induced from $H^*(S(G,X),\Lambda_V)_\pi$. This is the $p$-adic counterpart of the motivic rationality prediction.