Step 1. By Proposition 2.9(iv), we have

Hence replacing $\theta$ by $\theta+dd^{c}f$ and $f$ by $0$, we can assume that $f=0$ by Proposition 3.5(b).

Step 2. Let $\Delta\coloneqq S(\mathscr{X})$ denote the skeleton of $\mathscr{X}$. By Propositions 3.5 and 3.8, we get that

for the set $\mathcal{A}$ of non-positive piecewise linear functions $F$ on $\Delta$ such that $mF$ is integral $\Gamma$-affine for some $m\in\mathbb{N}_{>0}$ and such that $F\circ\tau$ is $\theta$-psh. Note that the piecewise linear functions are not assumed to be affine on the edges of $\Delta$. Since $X$ is a smooth projective curve and the de Rham class $\{\theta\}$ is nef, it is clear that $\mathcal{A}$ is non-empty. We introduce the function $F_{0}\colon\Delta\to\mathbb{R}$ defined by

By (3.2), we get that ${P}_{\theta}(0)=F_{0}\circ\tau$. Hence we can reduce (i) to prove that $F_{0}$ is affine on each edge of $\Delta$.

Step 3. For $F\in\mathcal{A}$, let $L(F)\colon\Delta\to\mathbb{R}$ be the function which is affine on the edges of $\Delta$ and which agrees with $F$ on the set $V$ of vertices of $\Delta$. As $F\leq 0$, we deduce immediately $L(F)\leq 0$. Since $F\circ\tau$ is $\theta$-psh, Proposition 3.7 shows that $F$ is convex on each edge of $\Delta$ and hence $F\leq L(F)$.

By passing from $F$ to $L(F)$, the slopes do not decrease in the vertices and using that $F\circ\tau$ is $\theta$-psh, it follows from Proposition 3.7 that $L(F)\circ\tau$ is $\theta$-psh as well.

The slopes of the function $L(F)$ might be non-rational. However, we can approximate the slopes of $L(F)$ in a rational way at any vertex and thus for any $\varepsilon>0$ we find a piecewise linear function $L_{\varepsilon}(F)$ on $\Delta$ such that

1. (i)

   $L_{\varepsilon}(F)$ agrees with $F$ on $V$.

2. (ii)

   $L_{\varepsilon}(F)$ has rational slopes.

3. (iii)

   $L_{\varepsilon}(F)$ is convex on the edges of $\Delta$.

4. (iv)

   $L_{\varepsilon}(F)\geq F$.

5. (v)

   $\sup|L_{\varepsilon}(F)-L(F)|<\varepsilon$

We claim that $L_{\varepsilon}(F)\in\mathcal{A}$. It follows from (ii) that $mL_{\varepsilon}(F)$ is integral $\Gamma$-affine for some $m\in\mathbb{N}_{>0}$. Since $F$ and $L(F)$ agree on $V$, it is clear from (i) and (iii) that $L_{\varepsilon}(F)\leq L(F)\leq 0$. To show that $L_{\varepsilon}(F)\circ\tau$ is $\theta$-psh, we use the slope criterion from Proposition 3.7. Note that (3.1) is fulfilled in the interior of each edge of $\Delta$ by (iii). In a vertex of $\Delta$, the inequality (3.1) is satisfied by using the corresponding inequality for $F$, (i) and (iv). This proves $L_{\varepsilon}(F)\in\mathcal{A}$. As a consequence we find

Step 4. We claim $F_{0}=L(F_{0})$. First note that since the $\max$ of convex functions in convex, $F_{0}$ is convex on each edge of $\Delta$, thus $F_{0}\leq L(F_{0})$.

We pick an $\varepsilon>0$. For any $v\in V$, there is $f_{v}\in\mathcal{A}$ with $f_{v}(v)>L(F_{0})(v)-\varepsilon$. It follows from [GM16, Prop. 3.12] that the maximum of two $\theta$-psh model functions is again a $\theta$-psh model function. Using 2.5, we conclude that $\mathcal{A}$ is closed under the operation $\max$. Thus $L(\max\{f_{v}\mid v\in V\})\in\mathcal{A}$ is in $\varepsilon$-distance to $L(F_{0})$ at every vertex of $\Delta$ and hence at every point of $\Delta$. As $\varepsilon>0$ can be chosen arbitrarily small, (3.3) yields $L(F_{0})\leq F_{0}$ and hence we get Step 4. Note that Step 4 proves (i).

Step 5. In the case $\Gamma\subset\mathbb{Q}$, we have $L(F)\in\mathcal{A}$ for $F\in\mathcal{A}$. Indeed, we note that in this special case the edges have rational lengths and $F$ takes rational values at $V$. We deduce from Proposition 3.5 that $L(F)\circ\tau$ is a model function. Let $\mathcal{B}$ be the set of $F\in\mathcal{A}$ such that $F$ is affine on every edge of $\Delta$. For $F\in\mathcal{A}$ we thus have $L(F)\in\mathcal{B}$ which shows via (3.3) that we might restrict the $\sup$ to $\mathcal{B}$ in the definition of $F_{0}$. Recall that $V$ is the set of vertices of $\Delta$. Since the edge lengths of $\Delta$ are rational, the map $\Psi\colon\mathcal{B}\to\mathbb{R}^{V}$ defined by $\Psi(F)=(F(v))_{v\in V}$ identifies $\mathcal{B}$ with the rational points of a rational polyhedron in $\mathbb{R}^{V}$ defined by the linear inequalities of Proposition 3.7. Note that by affineness on the edges, we need to check the slope inequalities only at the vertices. If a rational linear form $\varphi$ is bounded from above on a rational polyhedron $P$, then $\varphi|_{P}$ achieves its maximum in a rational point. Hence there exists $G\in\mathcal{B}$ such that

We claim that $G=F_{0}$. Considering $F\in\mathcal{B}$, we get $\max(G,F)\in\mathcal{A}$ by Step 4 and hence $H^{\prime}\coloneqq L(\max(G,F))\in\mathcal{B}$. Hence $H^{\prime}\geq G$, $H^{\prime}\geq F$ and $H^{\prime}\in\mathcal{B}$. But by (3.4) we deduce that for $v\in V$ we have $H^{\prime}(v)=G(v)$. Since functions in $\mathcal{B}$ are determined by their values on $V$, we have $H^{\prime}=G$. Hence $G\geq F$, whence $G=F_{0}$. It follows that $P_{\theta}(0)=F_{0}\circ\tau=G\circ\tau$ is a $\theta$-psh function proving (ii). ∎