  Recently it was proposed that, 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 \cite{Basu:2022crn}. In this paper, 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. We assume that the cutoff brane also plays the same role as the
KR brane in AdS/BCFT, hence the path-integral-optimized purification for the
interval is in the island phase. This explains the appearance of negative
mutual information observed in \cite{Camargo:2022mme}. We check that, the
entanglement entropy and the balanced partial entanglement entropy (BPE)
calculated via the island formulas, exactly match with the RT formula and the
entanglement wedge cross-section (EWCS), which are allowed to anchor on the
cutoff brane.
