Generation conjecture for Gorenstein kernels of tautological rings

For $g,n\geq 0$, let $d53c^{\mathrm{ct}}_{g,n}$ be the moduli space of stable $n$-pointed curves of compact type, let $d52d^*(d53c^{\mathrm{ct}}_{g,n})$ be its tautological ring, and let $d4aa_{g,n}$ be the Gorenstein kernel. For every $s\times n$ matrix $d538$, let $d6a5_{g,\mathsf A}$ be the associated Abel–Jacobi pullback defect class in $d52d^{g-1+s}(d53c^{\mathrm{ct}}_{g,n})$.

Generation conjecture for Gorenstein kernels. The following hold: (P1) $d6a5_{g,\mathsf A}\in\mathsf{K}_{g,n}$ for all $g$ and all $s\times n$ matrices $d538$; and (P2) the system of ideals $d4aa_{g,n}\subset\mathsf{R}^*(\mathcal{M}^{\mathrm{ct}}_{g,n})$ is the smallest system closed under properties (i)–(v) of the cited section and for which (P1) holds.

This conjecture proposes a complete geometric and structural description of the Gorenstein kernels. The stated properties are supported by the computations and known closure results, but the general assertion remains open.