Similarity problem for certain martingale uniform algebras

Let $A$ be a proper uniform algebra admitting a nontrivial bounded point derivation. Then for a certain uniform algebra $A_1$ (related to $A$ much as the algebra of Hardy martingales on $\mathbb T^\infty$ is related to the disk algebra) there exists a bounded but not completely bounded homomorphism $\varphi\colon A_1\to B(H)$.