The completely bounded unital contraction conjecture for operator algebras

Let $A\subset B(\mathcal H)$ be a unital operator algebra. Let $\theta:A\to M_n(\mathbb{C})$ be a unital completely bounded homomorphism, and let $\alpha:A\to A$ be a unital antilinear complete contraction. Define the linear map

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

Here $\|\phi\|_{cb}=\sup_k\|\phi^{(k)}\|$ denotes the completely bounded norm. Completely bounded conjecture. If $\|\theta_\alpha\|_{cb}\leq 1$, then

$$\|\theta\|_{cb}\leq 2.$$

This is the completely bounded refinement of the preceding conjecture and brings operator-algebraic and dilation-theoretic methods into the problem. It is presented as a conjecture in the paper and remains open.