The pullback correspondence for equivariant slice towers

Let $G$ be a finite group, let $N$ be a normal subgroup, and let $X$ be a genuine $G/N$-spectrum. The slice tower of $X$ is the filtration by slices in the equivariant stable homotopy category, and $\phi_N^{\ast}X$ denotes its pullback to a genuine $G$-spectrum.

Pullback slice-tower conjecture. The slice tower for a $G/N$-spectrum pulls back to the slice tower for the pullback $G$-spectrum, where the $k$-slice becomes the $\big((k+1)|N|-1\big)$-slice.

The preceding theorem proves the corresponding connectivity and coconnectivity bounds for pullbacks, and the corollary establishes the zero-slice case. The full slice-tower correspondence is presented as a deeper consequence and is not resolved in the supplied text.