Circuit-free permutations attaining the lower bound for reduced words

Let $w$ be a permutation. Write $C(w)$ and $B(w)$ for the sets used in the paper, and let $\operatorname{wid}(w)$ and $\operatorname{sup}(w)$ denote its width and support, respectively. The permutation is called circuit-free when it has no circuit in the relevant associated structure.

Circuit-free permutation conjecture. The permutation $w$ is circuit-free, and thus achieves the lower bound in Theorem, if and only if at least one of the following conditions is satisfied:

$$|C(w)|=1,$$ $$|B(w)|=1,$$ $$\operatorname{wid}(w)=2,$$ $$\operatorname{wid}(w)=\operatorname{sup}(w)=3.$$

This conjecturally reformulates the paper's characterization of permutations achieving the lower bound in terms of width and support. The supplied text does not indicate whether the conjecture has been proved or disproved.