Spontaneous emergence of solitons in affine dKdV

Let $n$ be chosen, let $u$ and $v$ be glides with fixed reduced words $u=s_{i_1}\cdots s_{i_l}$ and $v=s_{j_1}\cdots s_{j_m}$, and assume that $s_{j_1}\cdots s_{j_m}s_{i_1}\cdots s_{i_l}$ is reduced. Choose positive parameters for the initial carrier ${\bf z}_{-\infty}$ and vacuum ${\bf w}$ so that the parameters remain unchanged after their interaction, and take an initial state equal to the vacuum except at finitely many positions. For $1\leq h\leq l$, let $f_h:\mathbb Z\to\mathbb R_{>0}$ assign to the $j$-th state the parameter corresponding to $s_{i_h}$. Spontaneous emergence conjecture. For any choice of the data as above spontaneous emergence of solitons occurs for a generic choice of the initial data. The conjecture concerns the appearance of shape-preserving waves with asymptotically constant velocity, together with residual radiation, under repeated affine dKdV evolution; the source gives no resolution status.