Conjecture on the top codimension of the product-extended tautological ring

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$, and let $\mathsf{R}^*_{\mathrm{pr}}(\mathcal{A}_g)\subset \operatorname{CH}^*(\mathcal{A}_g)$ be the $\mathbb{Q}$-vector subspace generated by classes from products $\mathcal{A}_{g_1}\times\cdots\times\mathcal{A}_{g_\ell}$ with $g=\sum_i g_i$ and $g_i\geq 1$, together with arbitrary polynomials in the lambda classes on the factors. Here $\mathsf{R}^k_{\mathrm{pr}}(\mathcal{A}_g)$ denotes its codimension-$k$ part.

Top-codimension conjecture. For all $g\geq 1$,

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

The product-extended ring enlarges the usual tautological ring by including product loci and their tautological classes. The proposition preceding the conjecture establishes that it is a $\mathbb{Q}$-algebra, admits product pushforwards, vanishes above codimension $\binom{g}{2}$, and is strictly larger than the usual tautological ring for $g=6$; the conjecture asserts that its maximal possible codimension is nevertheless one-dimensional.