Generation conjecture for the polar linearised double shuffle Lie algebra

Let $\mathfrak{p}\mathfrak{ls}$ be the Lie algebra of solutions to the polar linearised double shuffle equations, and let $\mathfrak{p}\mathfrak{ls}_1$ denote its depth-one part.

Generation conjecture. The Lie algebra $\mathfrak{p}\mathfrak{ls}$ is generated by $\mathfrak{p}\mathfrak{ls}_1$.

The conjecture concerns generation of all polar linearised double-shuffle solutions from depth one. The source gives no resolution.