The proof of Properties (i)–(vi) in [BFJ16a, Prop. 8.2] works in our setup as well. Property (vii) is obvious for $t\in\mathbb{Q}_{>0}$ and an easy approximation argument then shows (vii) in general. We have seen that $\theta$-psh model functions are closed under $\max$ and hence the $\theta$-psh model functions $\varphi\leq u$ form a directed family. We conclude that (viii) follows from Dini’s Theorem for nets [Kel75, p. 239] and the definition of $P_{\theta}(u)$. ∎