The dimension formula for the comtrans operad

Let $\mathcal{CT}$ be the symmetric weight-graded comtrans operad, and let $\mathcal{CT}(w)$ denote its component of weight $w$. In the free comtrans algebra, the multilinear subspace of degree $2w+1$ is taken on $2w+1$ generators. The comtrans operad dimension formula.

$$\dim \mathcal{CT}(w) = \frac{(3w)!}{w!(3!)^w}\cdot 5^w \qquad (w\geq 0).$$

This formula gives the dimensions of the weight components of the comtrans operad, equivalently the dimensions of the corresponding multilinear subspaces in the free comtrans algebra. The supplied text does not indicate whether the assertion has been proved or remains open.