Fixed-point product formula conjecture for evacuation of Kreweras words

Let $n\geq 1$, and let $\operatorname{{Evac}}$ and $\operatorname{{Evac}}^{*}$ denote the two evacuation operations on Kreweras words of length $3n$. For a word $w$, consider the fixed-point conditions $\operatorname{{Evac}}^{*}(w)=w$ and $\operatorname{{Evac}}(w)=w$.

Evacuation fixed-point conjecture. For all $n\geq 1$, the number of Kreweras words of length $3n$ satisfying $\operatorname{{Evac}}^{*}(w)=w$ is

$$\frac{3^{\lfloor n/2 \rfloor} 4^{\lceil n/2 \rceil} \prod_{j=1}^{\lfloor n/2 \rfloor}(3j-1) \prod_{j=1}^{\lceil n/2 \rceil}(3j-2) }{(n+1)!}.$$

The number satisfying $\operatorname{{Evac}}(w)=w$ is the same when $n$ is even and is $0$ when $n$ is odd. These formulas provide product expressions for the evacuation fixed points, while their status beyond the conjecture is not established in the supplied text.