The slice decomposition conjecture for the quaternionic norm of real bordism

Let $Q_8$ be the quaternion group, let $C_2$ be its indicated subgroup, and let $N_{C_2}^{Q_8} MU\mathbb{R}$ denote the quaternionic norm of real bordism. For a subgroup $L$ of $Q_8$, write $\rho_L$ for the real regular representation of $L$, $H N_{C_2}^{L}\underline{\mathbb{Z}}$ for the corresponding Eilenberg–Mac Lane spectrum, and $P_n^n$ for the $n$-slice of a $Q_8$-equivariant spectrum. Slice decomposition conjecture. The slices of $N_{C_2}^{Q_8} MU\mathbb{R}$ are of the form

$$P_n^n N_{C_2}^{Q_8} MU\mathbb{R} \simeq \bigvee {Q_8}_+\wedge_L \Sigma^{i\rho_L} H N_{C_2}^{L}\underline{\mathbb{Z}},$$

for subgroups $L$ of $Q_8$. The conjecture is motivated by the Hill–Hopkins–Ravenel description of the slices of $N_{C_2}^{C_{2^n}} MU\mathbb{R}$, but the corresponding formula for the quaternionic norm is not established in the supplied text.