The AdS/BCFT correspondence can be simulated by a holographic Weyl transformed CFT$_2$, where the cut-off brane plays the role of the Karch-Randall (KR) brane. In this talk, we focus on the Weyl transformation that optimizes the path integral computation of the reduced density matrix for a single interval in a holographic CFT$_2$. When we take the limit that one of the endpoint of the interval goes to infinity (a half line), such a holographic Weyl transformed CFT$_2$ matches the AdS/BCFT configuration for a BCFT with one boundary. Without taking the limit, the induced cutoff brane becomes a circle passing through the two endpoints of the interval. If we take the scalar that characterizes the Weyl transformation to be dynamical, a AdS2 gravity was introduced to the region with non-trivial Weyl transformation. Hence, the  path-integral-optimized purification for the interval is in the island phase. This explains the appearance of negative mutual information is such states.