Stable equality conjecture for the Torelli lower central series and Y-filtration

Fix inclusions of surfaces $\Sigma_{0,1}\subset\Sigma_{1,1}\subset\Sigma_{2,1}\subset\cdots$. Regard the Torelli group $\mathcal{I}_{g,1}$ and the monoid of homology cylinders $\mathcal{C}_{g,1}$ inside their stable direct limits, and define

$$\Gamma^{\mathrm{stab}}_j\mathcal{I}_{g,1}:=\mathcal{I}_{g,1}\cap\Gamma_j\!\left(\varinjlim\mathcal{I}_{g,1}\right),$$

with the stabilized Y-filtration given by $Y_j^{\mathrm{stab}}\mathcal{C}_{g,1}:=\mathcal{C}_{g,1}\cap\varinjlim Y_j\mathcal{C}_{g,1}$. The claim concerns the intersection of the Y-filtration with the Torelli group.

Stable equality conjecture. For any $g\geq 0$ and all $j\geq 1$,

$$\Gamma^{\mathrm{stab}}_j\mathcal{I}_{g,1}=\mathcal{I}_{g,1}\cap Y_j\mathcal{C}_{g,1}.$$

The preceding proposition establishes stability of the Y-filtration itself, while equality with the stabilized lower central series remains conjectural. This is presented as an equivalent formulation of the stable injectivity conjecture.