Let $\mu$ be a singular cardinal of countable cofinality. Let $ADS_\mu$ be the assertion that there exists $\langle A_\alpha\subset \mu: \alpha<\mu^+\rangle$ such that for all $\beta<\mu^+$, there exists $F: \beta \to \mu$ such that $\langle A_\alpha\backslash F(\alpha): \alpha<\beta\rangle$ is pairwise disjoint. It is a well-known fact that $ADS_\mu$ implies the failure of $Refl^*([\mu^+]^{\omega})$, which says for any stationary subset $S$ of $[\mu^+]^\omega$, there exists $W\in [\mu^+]^{\aleph_1}$ with $\omega_1\subset W$ and $cf(\sup W)=\omega_1$ such that $S\cap [W]^{\omega}$ is stationary.

$Refl([\mu]^{\omega})$ is almost the same assertion as $Refl^*([\mu^+]^\omega)$ except we do not require $cf(\sup W)=\omega_1$ (so it's weaker). It is also known that these two principles in general are not equivalent.

My question is: does $ADS_\mu$ imply the failure of $Refl([\mu^+]^\omega)$?