The same-recording-tableau conjecture for steady-state time

For permutations $v,w\in S_n$, let $\operatorname{Q}(v)$ and $\operatorname{Q}(w)$ denote their recording tableaux under the Robinson–Schensted correspondence. The steady-state time is the number of box-ball-system moves required for a permutation to reach steady state. Same-recording-tableau conjecture. If

$$\operatorname{Q}(v)=\operatorname{Q}(w),$$

then $v$ and $w$ have the same steady-state time. The source reports that this conjecture was proved in a sequel to the paper.