Exponential dimension conjecture for faithful tropical representations of plactic monoids

Let $\mathbb{P}_n$ denote the plactic monoid on rank $n$, and let a tropical representation of $\mathbb{P}_n$ be faithful when it distinguishes all elements of $\mathbb{P}_n$. The dimension of a family of such representations is measured as a function of $n$. Exponential dimension conjecture. Any family of faithful tropical representations for $\mathbb{P}_n$ has dimension which grows exponentially, with base at least $2$, as a function of $n$. The paper's representation has dimension $2^n$, and the authors conjecture that pruning redundant blocks cannot significantly improve this exponential growth; no proof of the lower bound is given.