Dimension formula conjecture for multilinear right-commutative monomials

Let $FRC_n$ be the multilinear subspace of degree $n$ in the free right-commutative algebra on $n$ generators. An ordered basis of $FRC_n$ consists of the distinct right-commutative monomials in degree $n$, ordered first by association type and then lexicographically by the underlying permutation. For all $n\geq 1$, the dimension formula conjecture.

$$\dim FRC_n \stackrel{?}{=} \frac{n(2n{-}2)!}{2^{n-1}(n{-}1)!}.$$

The formula is motivated by computed dimensions for $1\leq n\leq 9$ and agrees with the cited Sloane sequence A001193. Its status is not established in the supplied source.