Finite-N fully quantum AEP lower bound

Let $\rho_{AB}$ be a bipartite state on finite-dimensional systems $A$ and $B$, let $0<\varepsilon<1$, let $\eta$ be the finite-AEP parameter associated to $\rho_{AB}$, and let $n\in\mathbb{N}$ be such that $\rho_{AB}^{\otimes n}$ is the i.i.d. state on $A^nB^n$. If $n\ge \frac85\log(2/\varepsilon^2)$, then the conditional entropy quantity $H$ satisfies\[\frac1n H_{\min}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}} \ge H(A|B)_{\rho}-\frac{\delta(\varepsilon,\eta)}{\sqrt n},\]where $\delta(\varepsilon,\eta):=4\log\eta\sqrt{\log(2/\varepsilon^2)}$.