The multiplicative lifting conjecture for the slice filtration

Let ${\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)$ be the category of symmetric $\mathbb{P}^1$-spectra over $k$, and let ${\operatorname{{\mathcal S}{\mathcal H}}}(k)$ be its homotopy category. Write $s_n$ for the $n$th slice functor and $\oplus_n s_n$ for the resulting graded functor. The multiplicative lifting conjecture. The functor $\oplus_n s_n$ and the natural transformation

$$\mu_{*,*}:\oplus_n s_n\wedge\oplus_n s_n\to\oplus_n s_n\circ\wedge$$

on ${\operatorname{{\mathcal S}{\mathcal H}}}(k)$ lift to a functor on ${\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)$ with values in graded objects, together with the same natural transformation as a natural transformation of bi-graded functors

$${\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)\otimes{\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k)\to{\mathbf{Spt}}^\Sigma_{{\mathbb P}^1}(k).$$

Such a lift would provide a multiplicative refinement of the slice filtration at the level of symmetric spectra. The authors explicitly call this a working assumption or conjecture, and the supplied text gives no resolution.