  We use numerical bootstrap techniques to study correlation functions of
traceless symmetric tensors of $O(N)$ with two indexes $t_{ij}$. We obtain
upper bounds on operator dimensions for all the relevant representations and
several values of $N$. We discover several families of kinks, which do not
correspond to any known model and we discuss possible candidates. We then
specialize to the case $N=4$, which has been conjectured to describe a phase
transition in the antiferromagnetic real projective model ARP$^{3}$. Lattice
simulations provide strong evidence for the existence of a second order phase
transition, while an effective field theory approach does not predict any fixed
point. We identify a set of assumptions that constrain operator dimensions to a
closed region overlapping with the lattice prediction. The region is still
present after pushing the numerics in the single correlator case or when
considering a mixed system involving $t$ and the lowest dimension scalar
singlet.
