Filtered synthetic deformation models motivic realizations

Let $\mathcal{C}$ be the realization target, let $L$ be its Tate object, and let $\nu(L^{\otimes *})$ be the filtered object induced by the slice tower. Write $\operatorname{Fil}(\mathcal{C})$ for filtered objects, $\operatorname{Syn}^{\operatorname{ev}}(\mathcal{C})$ for the synthetic deformation, and $\operatorname{Mod}_{\operatorname{fil}^{*}(L^{\otimes 0})}(\operatorname{Fil}(\mathcal{C}))$ for modules over $\operatorname{fil}^{*}(L^{\otimes 0})$. Filtered synthetic deformation conjecture. The following should hold: (1) the internal mapping object functor

$$\operatorname{Map}_{\mathcal{C}}(\nu(L^{\otimes *}),-):\operatorname{Syn}^{\operatorname{ev}}(\mathcal{C})\to\operatorname{Fil}(\mathcal{C})$$

can be promoted to an equivalence

$$\operatorname{Syn}^{\operatorname{ev}}(\mathcal{C})\simeq \operatorname{Mod}_{\operatorname{fil}^{*}(L^{\otimes 0})}(\operatorname{Fil}(\mathcal{C})),$$

where

$$\operatorname{fil}^{*}(L^{\otimes 0})\simeq\operatorname{End}_{\mathcal{C}}(\nu(L^{\otimes *}),\nu(L^{\otimes *})).$$

(2) Through this equivalence, the synthetic realization functor

$$\mathrm{Re}^{\operatorname{syn}}:\mathcal{SH}(k)\to\operatorname{Fil}(\mathcal{C})$$

can be identified on Artin--Tate objects with the functor sending $X\in\mathcal{SH}^{\operatorname{AT}}(k)$ to

$$\cdots\to\mathrm{Re}(\mathrm{f}_{1}X)\to\mathrm{Re}(\mathrm{f}_{0}X)\to\mathrm{Re}(\mathrm{f}_{-1}X)\to\cdots,$$

the realization of its tower of effective covers. This would relate the synthetic deformation to the filtered-object perspective on Adams filtrations and identify motivic realizations of Artin--Tate objects with realizations of their effective-cover towers.