Twisted associativity of the cyclically reduced product of words, part 2

The cyclically reduced product of two words $u, v$, denoted $u * v$, is the cyclically reduced form of the concatenation of $u$ by $v$. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations. In a previous paper we have proved that $*$ verifies generalizations of properties of the product in the free group. In another previous paper we have proved that $*$ verifies a generalized version of the associativity property in a special case. In the present paper we prove that a more general version of the associativity property holds for $*$ in the general case.