The finite-carrier conjecture for positive solitons of a6(n,k)

Let $\Phi(n,k)$ be the cellular automaton whose positive-soliton configurations evolve through states $\mathbf{Y}_i^t$ and carriers $\mathbf{Z}_i^t$. For $p=1,\ldots,n-1$, define

$$m_p=(\underbrace{1,\ldots,1}_{p-1},0,\underbrace{1,\ldots,1}_{n-p-1}),$$

and let $m_n=(1,1,\ldots,1)$. Set

$$M=\{m_1,\ldots,m_n\}\subset\{0,1\}^{n-1}.$$

Finite-carrier conjecture. When only positive solitons occur in $\Phi(n,k)$, the carriers $\mathbf{Z}_i^t$ take values in the finite set $M$. This is intended to express the observed duality between positive-soliton dynamics in $\Phi(n,k)$ and the states of the $\mathfrak{sl}_n$-box-ball system; the supplied text gives no resolution of the conjecture.