Motivic filtration conjecture for relative topological periodic homology

Let $R$ be an animated $\mathbf{Z}_p$-algebra, and let $\hat{{\mathbbl{\Delta}}}_{R/\mathbf{Z}_p[\\![{\widetilde{p}}]\\!]}$ be the Nygaard completion of ${{\widetilde{p}}\Omega}_R$. Let $\mathrm{F}^\star_\mathrm{mot}$ denote a motivic filtration.

Motivic filtration conjecture. The spectrum $\mathrm{TP}(R/X(p))$ admits a motivic filtration satisfying

$$\mathrm{gr}^i_\mathrm{mot}\mathrm{TP}(R/X(p))\simeq \hat{{\mathbbl{\Delta}}}_{R/\mathbf{Z}_p[\\![{\widetilde{p}}]\\!]}[2i]\otimes_R\epsilon^R.$$

This expectation is motivated by computations for polynomial algebras and predicts that relative topological periodic homology is governed by Nygaard-completed {{\widetilde{p}}-de Rham/prismatic cohomology; its status is not resolved in the supplied text.