Ranked-poset conjecture for tilted Bruhat order

Let $\mathbf{a}\in[n]^n$, and let $\leq_{\mathbf{a}}$ denote the $\mathbf{a}$-tilted Bruhat order on $S_n$.

Ranked-poset conjecture for tilted Bruhat order. The poset $(S_n,\leq_{\mathbf{a}})$ is ranked.

A rank function would provide basic structural information about the tilted Bruhat order. The source explicitly notes that the $\mathbf{a}$-tilted length $\ell_{\mathbf{a}}(w)$ does not serve as a rank function and gives no resolution of the conjecture.