The general sweep-map bijectivity conjecture

Let $A=x_1,\ldots,x_k\}$ be an alphabet and $\operatorname{wt}:A\rightarrow\mathbb{Z}$ a weight function. For nonnegative integers $n_1,\ldots,n_k$, let $\mathcal{R}^{\mathrm{word}}(x_1^{n_1}\cdots x_k^{n_k})$ be the words with the prescribed multiplicities, and let $\mathcal{D}^{\mathrm{word}}_{\operatorname{wt}}(x_1^{n_1}\cdots x_k^{n_k})$ be those words whose levels relative to $\operatorname{wt}$ are all nonnegative. Sweep-map bijectivity conjecture. The map $\operatorname{sw}_{\operatorname{wt}}$ maps each of these two sets bijectively to itself. This asserts that the general sweep map preserves both content and the nonnegative-level condition.