Compute the Turaev cobracket of the correction term

Compute the Turaev cobracket of the element j in the symplectic-invariant degree-six ideal component \mathfrak{j}_{g,1}(6)^{\mathrm{Sp}}, where j is the correction term appearing in the decomposition of the first Galois obstruction \sigma_3(g).

Background

The paper expresses the first Galois obstruction as $\sigma_3(g)=180(g-1)g\,\varepsilon-j$, where $\varepsilon$ lies in the kernel of the Enomoto–Satoh trace map and $j\in\mathfrak{j}_{g,1}(6)^{\mathrm{Sp}}$ is a correction term arising from the ideal that vanishes after passage to the closed-surface setting. The authors explicitly compute $ES_6(\sigma_3(g))=ES_6(-j)$, but they do not compute the Turaev cobracket of $j$.