If we pass to the completion $\mathbb{C}_{K}$ of an algebraic closure of $K$, there is a strictly semistable model $\mathscr{X}^{\prime}$ dominating $\mathscr{X}$ such that $f$ is determined on $\mathscr{X}$. This is proven in [BL85, §7]. We note that the property $\theta$-psh holds if and only if the corresponding property holds after base change to $\mathbb{C}_{K}$. This is a consequence of the projection formula in algebraic intersection theory. Since the degree is invariant under base change, it follows from [Thu05, Prop. 2.2.21] that the left hand side of (3.1) is invariant under base change as well. We conclude that we may assume that $K$ is algebraically closed and that $\mathscr{X}=\mathscr{X}^{\prime}$, i.e. $f$ is determined on $\mathscr{X}$. Then (3.1) follows from the slope formula of Katz, Rabinoff, and Zureick-Brown [KRZB16, Thm. 2.6]. ∎