The direct-summand conjecture for norm-preserving matrix powers

Let $A\in M_n$ be a complex matrix, and let $\|\cdot\|$ denote the operator norm. A matrix is an idempotent if $E^2=E$, and a scalar is unimodular if its absolute value is $1$.

Direct-summand conjecture. If

$$\|A^k\|=\|A\| \qquad\text{for all } k\geq 1,$$

then some power of $A$ has a direct summand of the form $\lambda E$, where $E$ is an idempotent and $|\lambda|=1$.

The paper states that this conjecture is disproved later in the work, so the claim is refuted rather than open.