Necessity of approximate Kakutani structure for irreducible weighted shifts

Let $T$ be an irreducible weighted shift with weights $\{\alpha_n\}_{n=1}^{\infty}$. For each $n\in\mathbb{N}$ and $\epsilon>0$, suppose there is an index $c_{n,\epsilon}\geq n$ such that

$$0<\alpha_{c_{n,\epsilon}}<\epsilon,$$

and

$$1\leq k\leq n\quad\Rightarrow\quad |\alpha_k-\alpha_{c_{n,\epsilon}-k}|<\epsilon.$$

Such a shift is called approximately Kakutani. The approximate Kakutani necessity conjecture. Every irreducible weighted shift in $\overline{\operatorname{CSO}}$ is approximately Kakutani. The preceding theorem proves that approximate Kakutani shifts belong to $\overline{\operatorname{CSO}}$; the conjecture asserts the converse, giving a characterization of irreducible weighted shifts in the closure of the complex symmetric operators. Its status is not resolved in the supplied source.