Constant-length enumeration conjecture for fully commutative elements in $G(m,m,3)$

Consider the complex reflection group $G(m,m,3)$ with its affine generating set, and let $\ell$ denote the length of an element. Constant-length enumeration conjecture. When $\ell>1$, the number of fully commutative elements of length $\ell$ is $6$. This is a pattern observed in the enumerative data for $G(m,m,3)$; no proof or resolution is given in the supplied source.