= Solution
At fixed $Q_T$, the pressure gradient is no longer fixed. As $p_s\to0$, the constitutive law gives a dilute particle fraction and hence $Q_p\to0$. Raising $p_s$ first adds particles and increases their flux. At high $p_s$, the packing becomes dense and nearly jammed; maintaining $Q_T$ then requires a large pressure gradient that sends most of the liquid through the packing by Darcy seepage, while $Q_p$ falls back toward zero. Thus $Q_p(p_s)$ has an interior maximum, and every particle flux below that maximum occurs at two solid pressures.
On the low-$p_s$ branch the suspension is dilute, shear is distributed broadly, particle motion carries much of the total volume, and the required $\Delta P$ is relatively small. On the high-$p_s$ branch the suspension contains a large nearly jammed plug, particles move slowly, seepage carries a substantial fraction of $Q_T$, and a much larger $\Delta P$ is required. Equal particle flux therefore does not imply equal concentration, flow structure, or pumping cost.
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