Noether–Lefschetz tautological ring socle and vanishing conjecture for abelian varieties

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$. Let $\mathsf{R}^*_{\mathrm{NL}}(\mathcal{A}_g)\subset \operatorname{CH}^*(\mathcal{A}_g)$ be the $\mathbb{Q}$-subalgebra generated by pushforwards of polynomials in Chern classes of automorphic algebraic vector bundles on marked irreducible components of Noether–Lefschetz loci. Noether–Lefschetz socle and vanishing conjecture. The ring satisfies

$$\mathsf{R}^{\binom{g}{2}}_{\mathrm{NL}}(\mathcal{A}_g)\cong \mathbb{Q}$$

and

$$\mathsf{R}^{k}_{\mathrm{NL}}(\mathcal{A}_g)=0\qquad\text{for }k>\binom{g}{2}.$$

This extends the proposed socle and vanishing behavior from the product-extended tautological ring to the larger ring generated by Noether–Lefschetz loci. The supplied passage gives no resolution or further evidence for these properties.