(Sketch) Convex functions on $(\mathbb{C}^{*})^{n}$ correspond to torus invariant psh functions via the log map (cf. Lemma 4.3 below). If $u$ is admissible, then near the toric boundary the appropriate local potential $u_{m}$ extends continuously over the boundary piece by the growth asymptote assumption and convexity, and the extension remains psh. Conversely, the asymptotic condition is dictated by the local boundedness of $u_{m}$ near the toric boundary pieces. ∎