Fill's characterization conjecture for minimizers of the adjacent-transposition spectral gap

Let $n\geq 3$, let $\vec{p}$ be a regular parameter vector, and let $K(\vec{p})$ be the transition matrix of the corresponding adjacent-transposition walk. Let $\lambda_{K(\vec{p})}$ be its spectral gap, let $\vec{p}_{\mathrm{unif}}$ be the uniform parameter vector, and write

$$\lambda_* = \lambda_{K(\vec{p}_{\mathrm{unif}})}.$$

A label $c\in[n]$ is neutral if

$$p_{c,i}=p_{i,c}=1/2\qquad\text{for all }i\neq c.$$

Fill's characterization conjecture. The following are equivalent:

1. $\lambda_{K(\vec{p})}=\lambda_*$.
2. $\vec{p}$ has a neutral label.

The implication from a neutral label to equality is straightforward, but the converse remains the main open content of the conjecture. The restriction $n\geq 3$ is necessary: for $n=2$, every regular parameter vector has the same spectral gap, whereas a neutral label need not exist.