Minimal-integrability conjecture for weak estimates of rough singular integral operators

Let $\Omega$ be homogeneous of degree zero on $\mathbb{R}^d$, satisfy the vanishing moment condition, and let $A$ be a function on $\mathbb{R}^d$ whose derivatives of order one belong to ${\rm BMO}(\mathbb{R}^d)$. The operators $T_{\Omega,A}$ and $\widetilde{T}_{\Omega,A}$ are the operators associated with $\Omega$ and $A$, and the weak $L\log L$ estimate for $T_{\Omega,A}$ and the weak $(1,1)$ estimate for $\widetilde{T}_{\Omega,A}$ are the estimates under consideration. Minimal-integrability conjecture. The condition

$$\Omega\in L(\log L)^{2}(\mathbb{S}^{d-1})$$

is minimal for these estimates, in the sense that the exponent $2$ cannot be replaced by any real number smaller than $2$. The paper presents this as the weakest possible condition for the stated weak-type results, but no proof or resolution is supplied in the given text.