  We combine supersymmetric localization results and the numerical conformal
bootstrap technique to study the 3d maximally supersymmetric (${\cal N} = 8$)
CFT on $N$ coincident M2-branes (the $U(N)_k \times U(N)_{-k}$ ABJM theory at
Chern-Simons level $k=1$). In particular, we perform a mixed correlator
bootstrap study of the superconformal primaries of the stress tensor multiplet
and of the next possible lowest-dimension half-BPS multiplet that is allowed by
3d ${\cal N} = 8$ superconformal symmetry. Of all known 3d ${\cal N} = 8$
SCFTs, the $k=1$ ABJM theory is the only one that contains both types of
multiplets in its operator spectrum. By imposing the values of the short OPE
coefficients that can be computed exactly using supersymmetric localization, we
are able to derive precise islands in the space of semi-short OPE coefficients
for an infinite number of such coefficients. We find that these islands
decrease in size with increasing $N$. More generally, we also analyze 3d ${\cal
N} = 8$ SCFT that contain both aforementioned multiplets in their operator
spectra without inputing any additional information that is specific to ABJM
theory. For such theories, we compute upper and lower bounds on the semi-short
OPE coefficients as well as upper bounds on the scaling dimension of the lowest
unprotected scalar operator. These latter bounds are more constraining than the
analogous bounds previously derived from a single correlator bootstrap of the
stress tensor multiplet. This leads us to conjecture that the $U(N)_2 \times
U(N+1)_{-2}$ ABJ theory, and not the $k=1$ ABJM theory, saturates the single
correlator bounds.
