Symmetry conjecture for fully commutative elements in rank-two Shephard groups

Let $a$ and $b$ be positive integers. In the notation $a[b]a$ for the rank-two Shephard group with endpoint parameters $a,a$ and braid parameter $b$, and $b[a]b$ for the group with these parameters exchanged, consider fully commutative elements having a unique reduced expression, enumerated by their length. Rank-two symmetry conjecture. The enumeration by length is identical for the groups $a[b]a$ and $b[a]b$. This conjecture arises from enumeration tables for exceptional Shephard groups; the supplied source gives no proof or resolution.