Equivalence of broken-line and seed convexity

Let $\varphi$ be a tropical function in a cluster-variety setting, and consider convexity along broken lines and convexity with respect to every seed. The convexity-equivalence conjecture. Convexity along broken lines is always equivalent to convexity with respect to every seed.

The notions are introduced for broader cluster-variety settings, and the paper notes that the equivalence is proved in dimension $2$ using a further wall-crossing statement, while the general assertion remains open.