By [Ber04, Theorem 5.1.1] we know that $|f(x)|\neq 0$ and that $\varphi$ is piecewise affine linear on $\Delta$. By [Ber99, Theorem 5.2] we have $\varphi\leq\varphi\circ p_{\mathfrak{X}}$. Assume there is a face $\tau$ of $\Delta$ on which $\varphi$ is not convex, i.e. there are $x,y\in\tau$ and $t\in(0,1)$ such that

$$\delta:=\varphi(tx+(1-t)y)-t\varphi(x)-(1-t)\varphi(y)>0.$$

By base change we can assume that $K$ is algebraically closed and then by density of the value group $\Gamma$ and continuity of $\varphi$ that the coordinates of $x$ and $y$ are in $\Gamma$. Choose a $\Gamma$-rational polytopal subdivision of $\Delta$ which only has $x$ and $y$ as additional vertices. By Construction 2.6 we get an admissible formal model $\mathfrak{X}^{\prime\prime}$ of $\mathfrak{X}^{\textup{an}}$ dominating $\mathfrak{X}$. Choose an affine open $U\subseteq\tilde{\mathfrak{X}}^{\prime\prime}$ which contains $\red(tx+(1-t)y)$. By the stratum face correspondence (Proposition 2.8 and Corollary 2.9) the vertices $x$ and $y$ correspond to irreducible components of $\tilde{\mathfrak{X}}^{\prime\prime}$. By taking out all other irreducible components we may assume that $U$ intersects only those corresponding to $x$ and $y$. Then $V:=\red^{-1}(U)$ is a strictly $K$-affinoid domain by [Bos77, Theorem 3.1]. By [Ber99, Proposition 1.4] its canonical reduction has two irreducible components, namely those corresponding to $x$ and $y$. Hence the Shilov boundary of $V$ is the set $\{x,y\}$ by [Ber90, Proposition 2.4.4] and we get $|f(tx+(1-t)y)|\leq\max\left\{|f(x)|,|f(y)|\right\}$. Since $x\neq y$, by restricting to a building block $\mathfrak{U}$, we can find a coordinate function $g\in\mathcal{O}(\mathfrak{U}^{\textup{an}})^{\times}$ such that $|g(x)|\neq|g(y)|$. Then we can find $N\in\mathbb{N}_{>0}$ and $m\in\mathbb{Z}$ such that

$$\Big|\log|f^{N}g^{m}(x)|-\log|f^{N}g^{m}(y)|\Big|=\Big|N(\varphi(x)-\varphi(y))+m(\log|g(x)|-\log|g(y)|)\Big|<N\delta.$$

Since $\log|g^{m}|$ is affine linear on $\tau$ we get

$$\log|f^{N}g^{m}(tx+(1-t)y)|-t\log|f^{N}g^{m}(x)|-(1-t)\log|f^{N}g^{m}(y)|=N\delta.$$

Hence by replacing $f$ with $f^{N}g^{m}$ and $\delta$ by $N\delta$ we can assume

$$\delta:=\varphi(tx+(1-t)y)-t\varphi(x)-(1-t)\varphi(y)>0.$$

and

$$|\varphi(x)-\varphi(y)|<\delta.$$

Then

$$\varphi(tx+(1-t)y)=\delta+t\varphi(x)+(1-t)\varphi(y)>t\varphi(x)+(1-t)\varphi(y)+|\varphi(x)-\varphi(y)|.$$

Now on the one hand we have

$$t\varphi(x)+(1-t)\varphi(y)+|\varphi(x)-\varphi(y)|\geq t\varphi(x)+(1-t)\varphi(y)+t(\varphi(y)-\varphi(x))=\varphi(y)$$

while on the other hand

$$t\varphi(x)+(1-t)\varphi(y)+|\varphi(x)-\varphi(y)|\geq\varphi(x)+t(\varphi(x)-\varphi(y)).$$

Together we get

$$\varphi(tx+(1-t)y)>\max\left\{\varphi(x),\varphi(y)\right\}.$$

But this violates our previous observation that $|f(tx+(1-t)y)|\leq\max\left\{|f(x)|,|f(y)|\right\}$. This finishes the proof. ∎

We may assume that $\|\cdot\|$ is a piecewise linear metric. Let $\mathscr{X}^{\prime}$ be an algebraic model of $X$ on which $(\mathcal{O}_{X^{\textup{an}}},\|\cdot\|)$ has an algebraic model $\mathscr{M}$. The section $1$ of $\mathcal{O}_{X}$ extends to a meromorphic section $s$ of $\mathscr{M}$ and then $\mathscr{M}=\mathcal{O}(D)$ for the vertical Cartier divisor $D=\Div(s)$ on $\mathscr{X}^{\prime}$. By [GW10, Theorem 13.98] we may assume that $\mathscr{X}^{\prime}$ is a vertical blowup of $\mathscr{X}$. Denote by $\pi$ the canonical map $\mathscr{X}^{\prime}\rightarrow\mathscr{X}$. We show first that $D$ is $\pi$-nef, i.e. $\deg(D\cdot C)\geq 0$ for any closed curve $C\subseteq\tilde{\mathscr{X}}^{\prime}$ which is contracted by $\pi$.
So let $x\in\tilde{\mathscr{X}}$ be a closed point and $C\subseteq\pi^{-1}(x)$ a curve. Then by the semipositivity assumption $\deg((\mathcal{O}(D)+\pi^{\ast}\mathscr{L})\cdot C)\geq 0$. But since $\pi_{\ast}(\pi^{\ast}\mathfrak{L}\cdot C)=\mathscr{L}\cdot\pi_{\ast}(C)=0$ we have $\deg(\pi^{\ast}\mathscr{L}\cdot C)=0$ and hence $\deg(D\cdot C)\geq 0$.
Now let $A$ be a $\pi$-ample vertical Cartier divisor on $\mathscr{X}^{\prime}$, e.g. $A=-E$ for the exceptional divisor $E$ of the blowup (this is $\pi$-ample by [GW10, Proposition 13.96]). Then $D+A$ is $\pi$-ample by the relative version of Kleiman’s criterion ([Deb01, Remark 7.41]). Furthermore, since $\mathcal{O}_{\mathscr{X}^{\prime}}(D)$ and $\mathcal{O}_{\mathscr{X}^{\prime}}(A)$ are coherent vertical fractional ideal sheaves, also $\mathfrak{a}:=\pi_{\ast}\mathcal{O}_{\mathscr{X}^{\prime}}(m(D+A))$ is a coherent vertical fractional ideal sheaf on $\mathscr{X}$ for any $m\in\mathbb{N}_{>0}$ by [Ull95, Theorem 5.3].
By the characterization of $\pi$-ampleness in [Gro61, Proposition 4.6.8] there exists some $m\in\mathbb{N}_{>0}$ such that $\pi^{\ast}\mathfrak{a}\rightarrow\mathcal{O}_{\mathscr{X}^{\prime}}(m(D+A))$ is surjective. This implies

$$\log|\mathfrak{a}|=\log|\pi^{\ast}\mathfrak{a}|=\log|\mathcal{O}_{\mathscr{X}^{\prime}}(m(D+A))|=m\cdot(\varphi-\log\|1\|_{\mathcal{O}_{\mathscr{X}^{\prime}}(A)})$$

and hence $\frac{1}{m}\log|\mathfrak{a}|=\varphi-\log\|1\|_{\mathcal{O}_{\mathscr{X}^{\prime}}(A)}$. Since we can replace $A$ by $\epsilon A$ for arbitrary small $\epsilon\in\mathbb{Q}_{>0}$ this concludes the proof. ∎

By Lemma B.3 we may approximate $\varphi$ by functions of the form $\frac{1}{d_{m}}\log|\mathfrak{a}_{m}|$ for some vertical coherent fractional ideals $\mathfrak{a}_{m}$ on $\mathscr{X}$. On the generic fibre of a building block $\mathfrak{U}$, the function $\log|\mathfrak{a}_{m}|$ is given as the maximum of the functions $\log|f|$ where $f$ runs through a finite set of generators of $\mathfrak{a}_{m}\Big|_{\mathfrak{U}}$. Since the properties we are looking for are stable under taking the maximum, these functions have them by Lemma B.1. But they are also stable under uniform limits so we are done. ∎