Failure of Rational Dilation on the Tetrablock via Parrott Homomorphisms

We determine the cotangent operator space structure at the origin of the tetrablock (denoted by $\mathbb{E}$). In particular we prove that ${\mathrm{COT}}_0(\bar{\mathbb{E}})=\mathrm{MIN}({\mathrm{\ell }}_1^2{\oplus }_{\mathrm{\infty }}\mathbb{C})$ completely isometrically. This shows the existence of commuting tuple of operators having $\bar{\mathbb{E}}$ as a spectral set but not as a complete spectral set.