Broken-line convexity of tropicalized regular functions

Let $Y$ be a log Calabi–Yau variety and let $f$ be a regular function on $Y$. Its tropicalization $f^{\operatorname{trop}}$ is the associated tropical function. Gross–Hacking–Keel–Kontsevich's conjecture. The tropicalization of any regular function on any log Calabi–Yau variety is convex along broken lines.

This is attributed to Gross, Hacking, Keel, and Kontsevich. In the paper, the conjecture is said to follow from the equivalence of broken-line and seed convexity, since globally regular functions restrict to regular functions on each seed torus.