Boundedness conjecture for the derivative-Hilbert operator on Hardy spaces

Let $H^p$ denote the Hardy space on the unit disk, let $\mu$ be a positive Borel measure on $[0,1)$, and let $\mathcal{DH}_\mu$ denote the derivative-Hilbert operator associated with $\mu$. Assume that $\mu$ is a $2$-Carleson measure. Derivative-Hilbert boundedness conjecture. Then $\mathcal{DH}_\mu$ is a bounded operator in $H^p$ for all $2<p<\infty$. The preceding result establishes the corresponding boundedness for $1\leq p\leq 2$; the conjecture asks whether the same conclusion extends to the remaining range $2<p<\infty$.