The free odd generation conjecture for the double shuffle Lie algebra

Let $\mathfrak{dm}$ be the double shuffle Lie algebra, graded by weight, and let $S=\{s_3,s_5,\ldots,s_{2n+1},\ldots\}$ contain one generator in each odd weight at least $3$.

Free odd generation conjecture. The double shuffle Lie algebra $\mathfrak{dm}$ is a free Lie algebra with exactly one generator in each odd weight $w\geq 3$:

$$\mathfrak{dm}\simeq\operatorname{Lie}(S).$$

This conjecture is motivated by conjectures of Deligne, Ihara and Drinfeld concerning Galois actions and the Grothendieck–Teichmüller Lie algebra. The latter is known to embed into $\mathfrak{dm}$, but the asserted freeness remains open.