Let $\|\cdot\|^{\prime}$ be any toric smooth metric. Then $\|\cdot\|_{\mathbb{S}}$ extends to a smooth metric if and only if $\log(\|s\|_{\mathbb{S}}/\|s\|^{\prime})$ can be extended to a smooth function on $X_{\Sigma}^{{\text{\rm an}}}$. But we have

$$\log(\|s\|_{\mathbb{S}}/\|s\|^{\prime})=\int_{\mathbb{S}^{{\text{\rm an}}}}\log(\|s(t\cdot p)\|/\|s(t\cdot p)\|^{\prime})\,\text{\rm d}\mu_{\operatorname{Haar}}(t)$$

and the right-hand side can be extended to a smooth function on the whole $X_{\Sigma}^{{\text{\rm an}}}$. Clearly the metric $\|\cdot\|_{\mathbb{S}}$ is toric. Moreover

$$c_{1}(L,\|\cdot\|_{\mathbb{S}})=\int_{\mathbb{S}^{{\text{\rm an}}}}t^{\ast}c_{1}(L,\|\cdot\|)\,\text{\rm d}\mu_{\operatorname{Haar}}(t).$$

Therefore, if $(L,\|\cdot\|)$ is semipositive, then $(L,\|\cdot\|_{\mathbb{S}})$ is semipositive. ∎