Let $C$ be $K$-proper curve in $\mathcal{X}_{K}$ and let $\mathcal{C}$ be its closure in $\mathcal{X}$. Since $\mathcal{C}$ is flat over $S$, the degree of $\mathcal{L}|_{\mathcal{C}}$ on the generic fiber and on the special fiber concide, which reads $\mathcal{L}\cdot C=\mathcal{L}\cdot\mathcal{C}_{0}$. Now $\mathcal{C}_{0}$ is an effective linear combination of vertical curves, and the result follows. ∎