Nonvanishing conjecture for Abel–Jacobi product-cycle defects

Let $g\geq 1$ and $s\geq 0$. Let $d53c^{\mathrm{ct}}_{g,2s}$ be the moduli space of stable $2s$-pointed curves of compact type, let $d52d^*(d53c^{\mathrm{ct}}_{g,2s})$ be its tautological ring, and let $d4aa_{g,2s}$ denote the Gorenstein kernel of its $d706_g$-pairing. For the Abel–Jacobi map and defect class, set $d6a5_{g,s}=\operatorname{aj}^*(\Delta_{g,s})\in\mathsf{R}^{g-1+s}(\mathcal{M}^{\mathrm{ct}}_{g,2s})$.

Nonvanishing conjecture for Abel–Jacobi product-cycle defects. The class $d6a5_{g,s}$ lies in the Gorenstein kernel and is nonzero for all but finitely many pairs of the form $(g\geq 2,s)$.

The kernel assertion is known in the relevant constructions, while the nonvanishing assertion is open for all $s\geq 0$ and is expected to explain failures of the $d706_g$-pairing to be perfect.