The multiplicative motivic perverse filtration conjecture for the Hitchin system

Let $h(M_{n,d})$ be the relative Chow motive of the Hitchin system over $B$, equipped with its motivic perverse filtration $P_\bullet h(M_{n,d})$. Let $\mathfrak p_k$ and $\mathfrak q_k$ denote the projector correspondences defining this filtration, and let $\Delta^{\mathrm{sm}}_{M_{n,d}/B}$ be the relative small diagonal in

$$M_{n,d}\times_BM_{n,d}\times_BM_{n,d}.$$

Multiplicative motivic perverse filtration conjecture. The motivic perverse filtration is multiplicative with respect to cup-product:

$$\cup:P_kh(M_{n,d})\times P_lh(M_{n,d})\to P_{k+l}h(M_{n,d}),$$

and equivalently

$$\mathfrak q_{k+l+1}\circ[\Delta^{\mathrm{sm}}_{M_{n,d}/B}]\circ(\mathfrak p_k\times\mathfrak p_l)=0$$

in $\operatorname{CH}_*(M_{n,d}\times_BM_{n,d}\times_BM_{n,d})$. This conjecture strengthens the sheaf-theoretic multiplicativity conjecture by expressing multiplicativity through tautological algebraic-cycle relations; its resolution is not supplied in the source.