Hästö–Diening characterization for the Cauchy singular integral on Carleson curves

Let $\Gamma$ be a simple Carleson curve, let $w:\Gamma\to[0,\infty]$ be a weight, and let $p:\Gamma\to(1,\infty)$ be continuous and satisfy the Dini–Lipschitz condition. Let $HD_{p(\cdot)}(\Gamma)$ denote the class of weights satisfying the Hästö–Diening condition. Hästö–Diening conjecture. The Cauchy singular integral operator $S$ is bounded on $L^{p(\cdot)}(\Gamma,w)$ if and only if $w\in HD_{p(\cdot)}(\Gamma)$. The condition is known to characterize boundedness of the Hardy–Littlewood maximal operator in the corresponding Euclidean setting, while its equivalence with boundedness of $S$ on Carleson curves is conjectured here.