  We perform a numerical bootstrap study of the mixed correlator system
containing the half-BPS operators of dimension two and three in $\mathcal N =
4$ Super Yang-Mills. This setup improves on previous works in the literature
that only considered single correlators of one or the other operator. We obtain
upper bounds on the leading twist in a given representation of the R-symmetry
by imposing gaps on the twist of all operators rather than the dimension of a
single one. As a result we find a tension between the large $N$ supergravity
predictions and the numerical finite $N$ results already at $N\sim 100$. A few
possible solutions are discussed: the extremal spectrum suggests that at large
but finite $N$, in addition to the double trace operators, there exists a
second tower of states with smaller dimension. We also obtain new bounds on the
dimension of operators which were not accessible with a single correlator
setup. Finally we consider bounds on the OPE coefficients of various operators.
The results obtained for the OPE coefficient of the lightest scalar singlet
show evidences of a two dimensional conformal manifold.
