Let $L$ be a toric line bundle on $X_{\Sigma}$ and let $\|\cdot\|$ be a metric on $L^{{\text{\rm an}}}$. If there is a toric section $s_{0}$ such that the function $p\mapsto\|s_{0}(p)\|$ is $\,\mathbb{S}^{{\text{\rm an}}}$-invariant, then, for every toric section $s$, the function $p\mapsto\|s(p)\|$ is $\,\mathbb{S}^{{\text{\rm an}}}$-invariant.