Motivic filtration conjecture for relative topological Hochschild homology

Let $R$ be an animated $\mathbf{Z}_p$-algebra. Write $J(p)$ for the quotient appearing in the relative topological Hochschild homology spectrum, let $\widehat{\Omega}^{\slashed{D}}_R$ denote the diffracted Hodge complex, and let $\mathrm{F}^\star_\mathrm{mot}$ and $\mathrm{F}^\mathrm{conj}_i$ denote the motivic and conjugate filtrations, respectively. Let $x$ denote the localization class.

Motivic filtration conjecture. There is a filtration $\mathrm{F}^\star_\mathrm{mot} \mathrm{THH}(R/J(p))$ such that

$$\mathrm{gr}^i_\mathrm{mot} \mathrm{THH}(R/J(p)) \simeq (\mathrm{F}^\mathrm{conj}_i \widehat{\Omega}^{\slashed{D}}_R)[2i],$$

$\Theta'_R: \mathrm{THH}(R/J(p))\shortrightarrow\Sigma^2\mathrm{THH}(R/J(p))$ preserves this filtration and induces $\Theta+i$ from the $i$th to the $(i-1)$st conjugate piece, and

$$\mathrm{gr}^i_\mathrm{mot}(\mathrm{THH}(R/J(p))[x^{-1}])\simeq\widehat{\Omega}^{\slashed{D}}_R[2i]$$

with localization inducing the inclusion of the conjugate filtration piece. This would identify the relative theory with a sheared Rees construction on the conjugate filtration; whether such a motivic filtration exists is the question posed here.