Enumerative conjecture for minimally-sorted permutations

Let $\mathfrak{M}_n$ be the set of minimally-sorted permutations under $s_{123,132}$. Enumerative conjecture. For every positive integer $n$,

$$\lvert\mathfrak{M}_{2n}\rvert=(n+1)\lvert\mathfrak{M}_{2n-1}\rvert.$$

This gives a recurrence between the numbers of minimally-sorted permutations of consecutive odd and even lengths; the source says it was computationally verified for $n\leq6$.