Weak monotonicity of exclusion-process mixing time in the number of particles

Let $G$ be an arbitrary graph with arbitrary transition rates, let $n$ be its number of vertices, and let $\mathrm{EX}(k)$ denote the exclusion process with $k$ particles. Write $t_{\mathrm{mix}}^{\mathrm{EX}(k)}$ for its mixing time. Weak monotonicity conjecture. There exists an absolute constant $C>0$ such that, whenever $k_1\leq k_2\leq n/2$,

$$t_{\mathrm{mix}}^{\mathrm{EX}(k_1)}\leq C t_{\mathrm{mix}}^{\mathrm{EX}(k_2)}.$$

Such monotonicity is immediate for independent particles but is not known for the exclusion process; establishing it would clarify how the mixing time depends on particle number without assumptions on the graph or rates.