The unital antilinear contraction conjecture for uniform algebras

Let $A$ be a uniform algebra, let $\theta:A\to M_n(\mathbb{C})$ be a continuous homomorphism, and let $\alpha:A\to A$ be a unital antilinear contraction, meaning that $\alpha(1)=1$. Define

$$\theta_\alpha(f)=\frac{1}{2}\bigl(\theta(f)+\theta(\alpha(f))^*\bigr) \quad (f\in A).$$

Unital uniform-algebra conjecture. If $\|\theta_\alpha\|\leq 1$, then

$$\|\theta\|\leq 2.$$

This conjecture would provide a homomorphism-theoretic route to the Crouzeix conjecture on numerical ranges, which asserts that $\|p(T)\|\leq 2\max_{z\in W(T)}|p(z)|$ for every polynomial $p$ and matrix $T$. The paper proves the bound $1+\sqrt{2}$ in the non-unital case and establishes the conjecture in two special cases, but the general assertion remains open.