This essentially follows from Claim 2.3.1, p. 39 of

J. Kollár, S. Kovács: Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200 (2013).

The Authors prove the result for $m=1$, but the same proof works without modifications for every $m \geq 1$. Compare also with Claim 2.3.2 (same page), Proposition 2.17 (p. 48) and Proposition 2.18 (p. 49).

The last result, applied with $S=\{ \text{point} \}$, $\Delta=\Delta_Y=\emptyset$, gives precisely the statement you need.

This essentially follows from Claim 2.3.1, p. 39 of

J. Kollár, S. Kovács: Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200 (2013).

The Authors prove the result for $m=1$, but the same proof works without modifications for every $m \geq 1$. Compare also with Claim 2.3.2 (same page), Proposition 2.17 (p. 48) and Proposition 2.18 (p. 49).

The last result, applied with $S=X, \, p_X=\text{id}_X$, $\Delta=\Delta_Y=\emptyset$, gives precisely the statement you need.

This essentially follows from Claim 2.3.1, p. 39 of

J. Kollar, S. Kovacs: Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200 (2013).

The Authors prove the result for $m=1$, but the same proof works without modifications for every $m \geq 1$. Compare also with Claim 2.3.2 (same page), Proposition 2.17 (p. 48) and Proposition 2.18 (p. 49).

The last result, applied with $S=\{ \text{point} \}$, $\Delta=\Delta_Y=\emptyset$, gives precisely the statement you need.

This essentially follows from Claim 2.3.1, p. 39 of

J. Kollár, S. Kovács: Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200 (2013).

The Authors prove the result for $m=1$, but the same proof works without modifications for every $m \geq 1$. Compare also with Claim 2.3.2 (same page), Proposition 2.17 (p. 48) and Proposition 2.18 (p. 49).

The last result, applied with $S=\{ \text{point} \}$, $\Delta=\Delta_Y=\emptyset$, gives precisely the statement you need.