  We study multiscalar theories with $\text{O}(N) \times \text{O}(2)$ symmetry.
These models have a stable fixed point in $d$ dimensions if $N$ is greater than
some critical value $N_c(d)$. Previous estimates of this critical value from
perturbative and non-perturbative renormalization group methods have produced
mutually incompatible results. We use numerical conformal bootstrap methods to
constrain $N_c(d)$ for $3 \leq d < 4$. Our results show that $N_c> 3.78$ for $d
= 3$. This favors the scenario that the physically relevant models with $N =
2,3$ in $d=3$ do not have a stable fixed point, indicating a first-order
transition. Our result exemplifies how conformal windows can be rigorously
constrained with modern numerical bootstrap algorithms.
