  We numerically study the phase structure of the CP(1) model in the presence
of a topological $\theta$-term, a regime afflicted by the sign problem for
conventional lattice Monte Carlo simulations. Using a bond-weighted tensor
renormalization group method, we compute the free energy for inverse couplings
ranging from $0\leq \beta \leq 1.1$ and find a CP-violating, first-order phase
transition at $\theta=\pi$. In contrast to previous findings, our numerical
results provide no evidence for a critical coupling $\beta_c<1.1$ above which a
second-order phase transition emerges at $\theta=\pi$ and/or the first-order
transition line bifurcates at $\theta\neq\pi$. If such a critical coupling
exists, as suggested by Haldane's conjecture, our study indicates that is
larger than $\beta_c>1.1$.
