Affine fully commutative element enumeration conjecture for $G(m,m,n)$

Let $a_m$ denote the number of fully commutative elements in $G(m,m,n)$, for $n\geq 3$, with the affine generating set. Affine enumeration conjecture. There is a positive integer $k$ such that

$$a_{m+1}=a_m+k.$$

In particular, $k=12$ when $n=3$ and $k=60$ when $n=4$. The conjecture is suggested by the enumerative data for $G(m,m,3)$ and $G(m,m,4)$; the general assertion and the specified constants remain unproved in the supplied source.