Esterle's Hilbert-space contraction conjecture

Let $E$ be a closed subset of the unit circle $\mathbb{T}$ of Lebesgue measure zero. A Hilbert-space contraction is a contraction acting on a Hilbert space, and $T$ is unitary when $T^*T=TT^*=I$. Esterle's conjecture. There exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a Hilbert-space contraction such that $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator.

This conjecture would extend the Banach-space result for strong $AA^+$-sets to Hilbert-space contractions with arbitrary closed null spectral sets. The paper proves the assertion under the additional hypothesis $\operatorname{rank}(I-T^*T)<\infty$, while the general case remains open.