Generalized Bloom characterization conjecture for iterated commutators

Let $T$ be a non-degenerate Calderón–Zygmund operator with Dini-continuous kernel, let $m\in\mathbb N$ and $p>1$, and let $\lambda$, $\mu$, and $\eta$ be arbitrary weights. Here $BMO_\eta$ denotes the weighted BMO space associated with $\eta$, and $T_b^m$ is the $m$-fold iterated commutator. Generalized Bloom characterization conjecture. If

$$b\in BMO_\eta\Rightarrow\|T_b^m\|_{L^p(\lambda)}\lesssim\|b\|_{BMO_\eta}^m\|f\|_{L^p(\mu)},$$

and

$$\|T_b^mf\|_{L^p(\lambda)}\lesssim\|f\|_{L^p(\mu)}\Rightarrow b\in BMO_\eta,$$

then $\lambda,\mu\in A_p$ and

$$\eta\simeq\left(\frac{\mu}{\lambda}\right)^{1/(pm)}.$$

The conjecture asks whether the usual Bloom-weight characterization remains valid for arbitrary weights; it is presented as a negation of that possibility and is unresolved in the source.