Fill's multiplicity conjecture for minimizing adjacent-transposition walks

Let $n\geq 3$ and let $\vec{p}$ be a regular parameter vector. A label is neutral if $p_{c,i}=p_{i,c}=1/2$ for every $i\neq c$. Let

$$N(\vec{p})=\mathopen{}\mathclose\bgroup\originalleft|\{i\in[n]:i\text{ is neutral}\}\aftergroup\egroup\originalright|$$

be the number of neutral labels, and let $M(\vec{p})$ be the algebraic multiplicity of $1-\lambda_*$ in $\operatorname{spec}(K(\vec{p}))$, where $\lambda_*$ is the uniform parameter vector's spectral gap. Fill's multiplicity conjecture. If $N(\vec{p})\notin\{n-2,n\}$, then

$$N(\vec{p})=M(\vec{p}).$$

If $N(\vec{p})\in\{n-2,n\}$, then

$$M(\vec{p})=n-1.$$

This conjecture concerns the multiplicity of the second largest eigenvalue in the conjectured equality cases and remains open in the supplied text.