By Proposition 3.77(2) and Proposition 3.80, the statements (1) and (2) are equivalent.

Let $\|\cdot\|$ be an approachable toric metric. By Corollary 5.17 the function $|\psi_{\|\cdot\|}-\Psi|$ is bounded. By approachability there is a sequence $\|\cdot\|_{l}$ of smooth (resp. algebraic) semipositive metrics that converges to $\|\cdot\|$. Since $\|\cdot\|$ is toric, $\|\cdot\|_{\mathbb{S}}=\|\cdot\|$. Hence, the sequence of toric metrics $(\|\cdot\|_{l})_{\mathbb{S}}$ also converges to $\|\cdot\|$. We denote $\psi_{l}=\psi_{(\|\cdot\|_{l})_{\mathbb{S}}}$. By Proposition 5.38 and Proposition 5.67 the functions $\psi_{l}$ are concave. Since the sequence $(\psi_{l})_{l}$ converge uniformly to $\psi_{\|\cdot\|}$, the latter is concave.

Let now $\psi$ be a concave function on $N_{\mathbb{R}}$ such that $|\Psi-\psi|$ is bounded. Then $\psi$ determines a metric $\|\cdot\|$ on the restriction of $L^{{\text{\rm an}}}$ to $X_{0}^{{\text{\rm an}}}$. Since $\operatorname{stab}(\psi)=\operatorname{stab}(\Psi)=\Delta_{\Psi}$, by Proposition 3.81 there is a sequence of rational piecewise affine concave functions $\psi_{l}$ that converge uniformly to $\psi$ and with $\operatorname{rec}(\psi_{l})=\Psi$. By Remark 5.46, the functions $\Psi-\psi_{l}$ can be extended to continuous functions on $N_{\Sigma}$. Therefore, $\Psi-\psi$ can be extended to a continuous function on $N_{\Sigma}$. Consequently the metric $\|\cdot\|$ can be extended to $X^{{\text{\rm an}}}$. Let $\|\cdot\|_{l}$ be the metric associated to $\psi_{l}$. Then the sequence of metrics $\|\cdot\|_{l}$ converges to $\|\cdot\|$. By Corollary 5.28, the metrics $\|\cdot\|_{l}$ are approachable. We deduce that $\|\cdot\|$ is approachable. ∎