Stable secondary-class polynomiality conjecture for the Johnson kernel

Let ${\cal K}_g$ be the Johnson kernel, let $d_i\in H^{4i-3}({\cal K}_g;{\mathbb Q})$ be the higher secondary classes, and let ${\cal M}_g$ act on $H^*({\cal K}_g;{\mathbb Q})$. Secondary-class conjecture. All classes $d_i$ are uniquely defined and non-trivial for sufficiently large genus, and

$$\lim_{g\to\infty}H^*({\cal K}_g;{\mathbb Q})^{{\cal M}_g}\cong {\mathbb Q}[d_1,d_2,d_3,\ldots].$$

This predicts that the stable mapping-class-group-invariant cohomology of the Johnson kernel is a polynomial algebra on the secondary classes; the source gives no resolution.