Square-free words obtained from prefixes by permutations

An infinite square-free word w over a three letter alphabet T is said to have a k-stem @s if w=@sw"1w"2... where for each i, there exists a permutation @p"i of T which extended to a morphism gives w"i=@p"i(@s). We show that there exists an infinite k-stem word for k=1,2,3,9 and 13@?k@?19, but not for 4@?k@?8 and 10@?k@?12. The problem whether k-stem words exist for each k>=20 remains open.