The maximum-rank-class and majority conjectures for idempotents in Jones monoids

Let $\mathcal{J}_n$ be the Jones monoid, let $E_n$ be its set of idempotents, let $\operatorname{rank}(\alpha)$ denote the rank of $\alpha\in E_n$, let $e_n=|E_n|$, and let $d_n^*$ denote the size of the largest rank class of idempotents. Maximum-rank-class and majority conjectures. For each $n>1$,

$$d_n^*=\max_r\left|\{\alpha\in E_n:\operatorname{rank}(\alpha)=r\}\right|$$

occuring at $r=1$ when $n$ is odd and at $r=2$ when $n$ is even; moreover, for $n\geq 4$,

$$2d_n^*>e_n.$$

The first assertion specifies which rank class contributes the most idempotents, while the second says that this largest class contains more than half of all idempotents. The claims are suggested by the tabulated values through $n=12$.