Vacuum structure of $\mathrm{SU}(2)$ Yang-Mills theory at $\theta = \pi$

In $4\mathrm{D}$ $\mathrm{SU}(N)$ Yang-Mills with topological angle $\theta$, the mixed anomaly between the $Z_{N}$ one-form center symmetry and CP at $\theta = \pi$ (Gaiotto, Kapustin, Komargodski, Seiberg 2017) forbids a trivially gapped confining vacuum. For $N = 2$, determine whether the vacuum at $\theta = \pi$ breaks CP spontaneously (two degenerate vacua and a first-order jump of the topological charge density), is deconfined (including gapped with topological order), or is gapless. Large-N arguments favor CP breaking.

A sign-problem-free computation (tensor network, quantum simulation, or controlled continuation from imaginary $\theta$) showing whether the topological charge density jumps at $\theta = \pi$ in the continuum limit.