Hydrodynamic speed convergence for the liquid bin model

Let $p_1+\dots+p_N=1$, let $a_1,\ldots,a_N>0$, and for $s\geq 1/a_1$ let $X^{(s)}$ be the infinite bin model with move distribution $\mu^{(s)}=\sum_{i=1}^N p_i\delta_{\lfloor s a_i\rfloor}$. Assume that the rescaled initial configurations $X^{(s)}(0)/s$ converge to a liquid-bin configuration $x(0)$, so that the rescaled processes $X^{(s)}(\lfloor st\rfloor)/s$ converge to the liquid bin model with parameters $a_1,\ldots,a_N,p_1,\ldots,p_N$. Speed convergence conjecture. As $s$ goes to infinity, the speed of the rescaled infinite bin model $X^{(s)}(\lfloor st\rfloor)$ converges to the speed of the liquid bin model with parameters $a_1,\ldots,a_N,p_1,\ldots,p_N$. The hydrodynamic-limit theorem establishes convergence of the trajectories on compact time intervals, but speed convergence is identified as an important missing property and remains open.