Gubler and Martin [22, section 3] argue that the a priori global notion of semipositivity can be localized on the Berkovich space, but their notion is quite abstract. It would be attractive to formulate a notion of convexity for functions on $\Delta_{\mathcal{X}}$ that precisely characterize the restrictions of continuous $\theta$-psh functions from $X_{K}^{an}$ to $\Delta_{\mathcal{X}}$. Compare also [12, Chapter 5] for another strategy to define plurisubharmonicity on the Berkovich space in terms of positivity of currents.