The two-dimensional shuffle-generator conjecture for conjugation invariants

Let $\mathsf{ConjInv}$ be the algebra of conjugation invariants, and call a homogeneous element a generator at level $n$ if it is one of the homogeneous generators used for this shuffle algebra. A shuffle with letters means a shuffle product involving one or more letters. Two-dimensional shuffle-generator conjecture. In dimension $d=2$, for every level $n\geq 2$, no homogeneous generator of the conjugation invariants is a shuffle with letters. Equivalently, the generators of the conjugation invariants are generators of the letter-reduced conjugation invariants; any conjugation invariant that is a shuffle with letters is a shuffle of conjugation invariants with letters. The claim concerns the structure of shuffle-algebra generators and is supported in the source by the displayed low-dimensional computations.