Almost strong compactness of $\kappa$ can be characterized as follows: for every $\delta < \kappa < \lambda$, there is an elementary embedding $j_{\delta,\lambda}: V \rightarrow M$ with critical point $\geq \delta$, so that $j_{\delta,\lambda}`` \lambda \subseteq D \in M$ and $M \vDash  |D|< j_{\delta,\lambda}(\kappa)$. Boney and Brook-Taylor were then wondering whether almost strong compactness is essentially the same as strong compactness. Recently, Goldberg has proved that if $\kappa$ is of uncountable cofinality then SCH from below implies these two closely related concepts are the same. In this joint work with Zhixing You, we show that these two can be different in general cases.