The positive-soliton duality conjecture for a6$n,k$ and the \mathfrak{sl}_n box-ball system

Let $\Phi(n,k)$ have initial configuration $(\mathbf{Y}_i^0)_i$ consisting only of positive solitons. Interchange space and time so that $\mathbf{Y}_i^t$ and $\mathbf{Z}_i^t$ are regarded respectively as a carrier and a state at time $i$ and space $t$. Let

$$\tilde{B}_1=\{(x_2,\ldots,x_n)\in\mathbb{Z}_{\geq0}^{n-1}:\sum_{i=2}^n x_i\leq1\},$$

and identify the carrier set $M$ with $\tilde{B}_1$ through $\beta_k$. Positive-soliton duality conjecture. Via $\beta_k$, the dynamics of the $\mathbf{Z}_i^t$ is identified with that of the $\mathfrak{sl}_n$-box-ball system, with $(0,\ldots,0)$ denoting an empty box and $(\underbrace{0,\ldots,0}_{p-2},1,\underbrace{0,\ldots,0}_{n-p})$ denoting a box containing a $p$-ball for $p=2,\ldots,n$. If $(\mathbf{Y}_i^t)_i$ includes a minimal-form soliton $(b_1,\ldots,b_{n-1})$ with $b_{k+1}=1$, the corresponding box-ball configuration contains

$$\underbrace{n\ldots n}_{b_1}\underbrace{n-1\ldots n-1}_{b_2}\underbrace{n-2\ldots n-2}_{b_3}\ldots\underbrace{2\ldots2}_{b_{n-1}},$$

whose velocity is $\sum_{p=1}^{n-1}b_p$. The conjecture gives a precise space-time duality and transfers soliton data, including velocity, between the two systems; no resolution is supplied in the text.