Linear weighted bounds for the Beurling-type operators $B^m$

Let $B^m$ denote the $m$th operator in the family considered in the paper, and let $w\in A_2$ with classical $A_2$ characteristic $[w]_{A_2}$. Linear $A_2$ conjecture for $B^m$. For every $w\in A_2$, one has

$$\|B^mf\|_{L^2(w)}\le C\,m\cdot [w]_{A_2}\|f\|_{L^2(w)}.$$

This is the specialization of the preceding rough-singular-integral conjecture to $B^m$. The stated bound removes the additional logarithmic or weight-dependent factor from the proven estimates, while retaining the linear dependence on $m$; its resolution is not specified in the source.