Clouâtre–Ostermann–Ransford conjecture on contractive homomorphisms

Let ${\mathcal A}$ be a uniform algebra, and let $\alpha:{\mathcal A}\to{\mathcal A}$ be a unital anti-linear contraction. Let $\theta:{\mathcal A}\to{\mathcal M}_n(\mathbb C)$ be a continuous unital algebra homomorphism, and define the linear map

$$\Lambda:{\mathcal A}\to{\mathcal M}_n(\mathbb C),\qquad f\mapsto\theta(f)+\theta(\alpha(f))^*.$$

Clouâtre–Ostermann–Ransford conjecture. (i) If $\|\Lambda\|\leq 2$, then $\|\theta\|\leq 2$. (ii) If $\alpha$ is a complete contraction, $\theta$ is completely bounded, and $\|\Lambda\|_{cb}\leq 2$, then $\|\theta\|_{cb}\leq 2$.

The source states that this conjecture was motivated by proofs related to the Crouzeix conjecture and that, if true, it would imply the Crouzeix conjecture. The supplied text gives no resolution status.