The log_p-conjecture for chromatic inclusions in equivariant stable homotopy theory

Let $G$ be a group of order $p^r$, where $p$ is prime, and let $\cat P_G(K,p,n)$ denote the equivariant prime associated with a subgroup $K\leq G$, the prime $p$, and chromatic height $n$. The conjecture concerns the possible inclusions between these primes.

The $\log_p$-conjecture. For all $n\geq r$, one has

$$\cat P_G(1,p,n)\not\subseteq \cat P_G(G,p,n-r+1).$$

Equivalently, the known inclusion with chromatic shift $r=\log_p(|G|)$ is sharp: there is no inclusion with the smaller shift $r-1$. This conjecture would remove the remaining indeterminacy in the topology of the spectrum of the equivariant stable homotopy category for arbitrary finite groups; it is known when the group has square-free order, while the general case remains open.