Stability of motivic $p$-completion under connective covers

Let $X\in\operatorname{Spc}(k)_*$ be a pointed motivic space, where $\operatorname{Spc}(k)_*$ denotes pointed motivic spaces, and let $\tau_{\geq 1}X$ denote its connective cover.

Connective-cover $p$-completion conjecture. If $X$ is $p$-complete, then $\tau_{\geq 1}X$ is also $p$-complete.

This would allow one to work with the principal refinement of the Postnikov tower in connected motivic spaces and identify the $p$-completion of a nilpotent motivic space with the $p$-completion of its underlying Nisnevich sheaf. The conjecture is presented as needed because motivic spaces do not form an $\infty$-topos, so the preservation of $p$-completeness under connective covers does not follow from the corresponding general arguments.