The Kuhn-Lloyd question: does the chromatic fixed-point loss of a p-group equal the shortest cyclic subnormal chain length?

Theorem 1.1: for every prime $p$, finite $p$-group $G$, subgroup $H$ and $n\ge0$, $r_n(G,H)=\ell(G,H)$, the shortest cyclic subnormal chain length; sharpness is witnessed by finite spectra, and the case $n=0$ includes rational detection. Combined with Balmer-Sanders reductions, a corollary determines the finite-height prime inclusions in the Balmer spectrum of finite genuine $G$-spectra for every finite group $G$. Not shown: anything about infinite height beyond what Balmer-Sanders already give, or non-finite spectra.

Theorem 1.1: for every prime $p$, finite $p$-group $G$, subgroup $H$ and $n\ge0$, $r_n(G,H)=\ell(G,H)$, the shortest cyclic subnormal chain length; sharpness is witnessed by finite spectra, and the case $n=0$ includes rational detection. Combined with Balmer-Sanders reductions, a corollary determines the finite-height prime inclusions in the Balmer spectrum of finite genuine $G$-spectra for every finite group $G$. Not shown: anything about infinite height beyond what Balmer-Sanders already give, or non-finite spectra.