Liu–Lou–Zhu's Carleson embedding conjecture for ${\mathcal Q}_s$

Let $0<s<1$ and let $\mu$ be a positive Borel measure on the open unit disc $\mathbb D$. The spaces ${\mathcal Q}_s$ and ${\mathcal T}_{s,2}^2(\mu)$ are linked by the identity map $id$. Liu–Lou–Zhu's conjecture. The map

$$id:{\mathcal Q}_s\mapsto {\mathcal T}_{s,2}^2(\mu)$$

is bounded if and only if $\mu$ is an $s$-Carleson measure. The paper states that a tiny-perturbed version of this conjecture is proved, while the exact endpoint assertion remains unresolved in the supplied text.