Soliton separation conjecture for affine general linear Lie superalgebra cellular automata

Let $p$ be any state, and let $T_{\infty}$ be the time-evolution map. A soliton separation conjecture asserts that there exist a positive integer $\widetilde{t}$, solitons $v_1,\ldots,v_D$ with respective speeds $d_1,\ldots,d_D$, and positive integers $c_1,\ldots,c_D$ such that, for every $t>\widetilde{t}$,

$$(T_{\infty})^t(p)=u_1^{\otimes(c_1+d_1(t-\widetilde{t}))}\otimes v_1\otimes u_1^{\otimes(c_2+(d_2-d_1)(t-\widetilde{t}))}\otimes v_2\otimes\cdots\otimes u_1^{\otimes(c_D+(d_D-d_{D-1})(t-\widetilde{t}))}\otimes v_D\otimes u_1^{\otimes\infty}.$$

Thus every state should eventually separate into solitons whose positions evolve linearly with time. The paper reports verification for all $U_q(\widehat{\mathfrak{gl}}(2|2))$ and $U_q(\widehat{\mathfrak{gl}}(3|3))$ states with at most five non-vacuum elements, while the general assertion remains open.