Holmes–Plummer enumeration conjecture for cds-sortable permutations

Let $\mathfrak{S}_n$ denote the symmetric group on $n$ elements, and let a permutation be cds-sortable if it can be sorted using the context-directed reversal operations described in the paper. For every integer $k\geq 0$, the number of cds-sortable elements of $\mathfrak{S}_{2k+1}$ is

$$\#\{\text{cds-sortable elements in }\mathfrak{S}_{2k+1}\}=(k+1)(2k)!.$$

Holmes–Plummer conjecture. The number of cds-sortable elements in $\mathfrak{S}_{2k+1}$ equals $(k+1)(2k)!$. This conjecture proposes a closed formula for the odd-degree terms in the enumeration of cds-sortable permutations; the paper states that a general formula was not known and attributes this conjecture to E. Holmes and P. A. Plummer.