Annihilating ideals and Agler--McCarthy spectral varieties in the bidisc

The closed unit bidisc $\overline{\mathbb{D}}^2$ is known to be a spectral set for any pair $(T_1,T_2)$ of commuting contractions. When each $T_i$ is pure and has finite defect, the pair admits a much smaller spectral set: the closure of a distinguished variety $V\subset \mathbb{D}^2$. We exhibit co