Surjectivity conjecture for the elliptic and polar linearised double shuffle Lie algebras

Let $\mathfrak{u}^{\varepsilon}$ be the bigraded Lie algebra generated by the elements $check{\varepsilon}_{2n}$, and let $\mathfrak{p}\mathfrak{ls}$ be the polar linearised double shuffle Lie algebra. There is an injective map

$$\mathfrak{u}^{\varepsilon}longrightarrowmathfrak{p}\mathfrak{ls}$$

which is an isomorphism in depth one:

$$\mathfrak{u}^{\varepsilon}_1congmathfrak{p}\mathfrak{ls}_1.$$

Surjectivity conjecture. The map is an isomorphism:

$$\mathfrak{u}^{\varepsilon}congmathfrak{p}\mathfrak{ls}.$$

The paper studies this elliptic analogue of Zagier's conjecture and proves it in depths at most three and in a stable limit, while the general statement remains open.