Pau–Zhao's generalized Carleson embedding conjecture for $F(p,p\alpha-2,s)$

Let $p>1$, let $\alpha>1$, let $0<s<1$, let $\mu$ be a positive Borel measure on the open unit disc $\mathbb D$, and let $F(p,p\alpha-2,s)$ and ${\mathfrak T}_{s,p}^{\infty}(\mu)$ denote the spaces in the source. Pau–Zhao's generalized conjecture. The identity map

$$id:F(p,p\alpha-2,s)\mapsto {\mathfrak T}_{s,p}^{\infty}(\mu)$$

is bounded if and only if $\mu$ is an $[s+p(\alpha-1)]$-Carleson measure. The source presents this as the analogue for $\alpha>1$ and reports results only for a different near-endpoint embedding, so the asserted equivalence remains open in the supplied text.