The stable lower central series conjecture for the Torelli group

Let $\Sigma_{g,1}$ be a compact oriented surface with one boundary component, let $\mathcal{I}_{g,1}$ be its Torelli group, and let $\mathcal{IC}_{g,1}$ be the monoid of homology cylinders over it. Let $\mathbf{c}:\mathcal{I}_{g,1}\to\mathcal{IC}_{g,1}$ be the mapping-cylinder map. For $k\ge 1$, write $\Gamma_k^{\operatorname{stab}}\mathcal{I}_{g,1}$ for the stable lower central series defined using the direct limit of Torelli groups.

Stable lower central series conjecture. The lower central series of the Torelli group stably coincides with the restriction of the $Y$-filtration:

$$\forall k\geq 1,\quad \Gamma_k^{\operatorname{stab}}\mathcal{I}_{g,1}=\mathbf{c}^{-1}\left(Y_k\mathcal{IC}_{g,1}\right).$$

This conjecture places the stable lower central series exactly at the level of the $Y$-filtration, refining the inclusions between the lower central, $Y$-, and Johnson filtrations. The supplied text gives no resolution status.