I am reading "The dual complex of singularities" by de Fernex, Kollár and Xu and in the proof of Corollary 24 I have encountered a bit of reasoning that confuses me.

Let $(X, \Delta)$ be a $\mathbb{Q}$-factorial pair and let $0 \in X$ be a point such that $X$ is Kawamata log terminal. Let $p: Y \to X$ be a projective birational morphism such that its restriction to $X \setminus \{0\}$ is an isomorphism. Let $E$ be the excepctional locus of $p$ and assume that the pair $(Y, E + p_*^{-1} \Delta)$ is divisorially log terminal.

One of the claims of the Corollary 24 is that if $(X, \Delta)$ is Kawamata log terminal then the dual intersection complex of $E$ (a certain cellular complex associated to the divisor) of contractible, though my question is not about this conclusion really.

The main construction of the proof is to run $(Y, E + p_*^{-1}\Delta)$-MMP over $X$ to get a sequence of pairs $(Y,E)=(Y_0, E_0), \ldots, (Y_i, E_i), \ldots$ (call the projection $\pi_i$). As far as I understand such an MMP terminates with a pair $(Y_r, E_r)$ such that $K_{Y_r} + E_r$ is $\pi_r$-nef, but the proof of Corollary 24 claims that if $(X, \Delta)$ is supposed to be Kawamata log terminal then the MMP actually terminates with $X$ (not sure what the boundary is supposed to be in this case).

I am only starting to get to grips with the minimal model program techniques and would really appreciate it if someone could explain how the singularities of $(X, \Delta)$ are connected with the assertion about what the pair the MMP terminates with is.

Making this into a concrete **question**: if $(Y_i, E_i)$ is a pair as
above, $Y_i \neq X$ and $(X, \Delta)$ is klt how does one see that
$K_{Y_i} + E_i$ is not $\pi_i$-nef and so the MMP can continue to run?