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 σ if w=σw1w2⋯ where for each i, there exists a permutation πi of T which extended to a morphism gives wi=πi(σ). 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.