Sufficiency of weak Hilbert-transform estimates in the absolutely continuous case

Let $A$ be the indefinite Sturm–Liouville operator, assume

$$\sigma_{\operatorname{disc}}(A)=\emptyset,$$

and suppose that both spectral measures $d\Sigma_+$ and $d\Sigma_-$ are absolutely continuous, so that $\Sigma_\pm=\Sigma_{\operatorname{ac}\pm}$. Let and denote the corresponding weak two-weight estimates for the Hilbert transform.

Absolutely continuous sufficiency conjecture. Under these assumptions, the conditions and are sufficient for $A$ to be similar to a selfadjoint operator.

This is a special-case sufficiency claim stated after the broader conjecture. The supplied material gives no resolution status, so it is recorded as open.