Generalized partial-estimate conjecture for operator-valued inner functions

Let $E$ be a closed subset of the unit circle $\mathbb{T}$ such that $|E|=0$. Let $\mathcal{F}$ and $\mathcal{F}'$ be separable Hilbert spaces, and let $\Theta:(\mathbb{C}_\infty\setminus E)\to\mathcal{L}(\mathcal{F},\mathcal{F}')$ be a holomorphic function whose restriction to the unit disk is an inner function, with $\Theta(\lambda)$ purely contractive for every $\lambda\in\mathbb{D}$. Generalized partial-estimate conjecture. There exists a positive sequence $\epsilon_n\to0$ such that

$$\liminf_{n\to\infty}\frac{\delta_n(\Theta)}{\epsilon_n}=0.$$

Here $\delta_n(\Theta)$ is the quantity defined in the paper's partial-estimate theorem. This conjecture would remove the finite-rank hypothesis from the main theorem and thereby imply Esterle's Hilbert-space contraction conjecture. The paper does not establish it in the general operator-valued setting.