  In this paper, we investigate equivariant quotients of the Real bordism
spectrum's multiplicative norm $MU^{((C_{2^n}))}$ by permutation summands.
These quotients are of interest because of their close relationship with higher
real $K$-theories. We introduce new techniques for computing the equivariant
homotopy groups of such quotients.
  As a new example, we examine the theories $BP^{((C_{2^n}))}\langle
m,m\rangle$. These spectra serve as natural equivariant generalizations of
connective integral Morava $K$-theories. We provide a complete computation of
the $a_{\sigma}$-localized slice spectral sequence of
$i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$, where $\sigma$ is the
real sign representation of $C_{2^{n-1}}$. To achieve this computation, we
establish a correspondence between this localized slice spectral sequence and
the $H\mathbb{F}_2$-based Adams spectral sequence in the category of
$H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. Furthermore, we provide a full
computation of the $a_{\lambda}$-localized slice spectral sequence of the
height-4 theory $BP^{((C_{4}))}\langle 2,2\rangle$. The $C_4$-slice spectral
sequence can be entirely recovered from this computation.
