Martin--Vial's Bloch--Beilinson criterion for sums of points

Let $T$ be a smooth projective variety whose algebra of holomorphic forms is generated in degree at most $d$. Let $F^\bullet$ be the conjectural Bloch--Beilinson filtration on $\operatorname{CH}_0(T)$, and let $x_1,\ldots,x_m,y_1,\ldots,y_m\in T$.

Martin--Vial's conjecture.

$$\sum_{i=1}^m[x_i]=\sum_{i=1}^m[y_i]\text{ in }\operatorname{CH}_0(T)$$

if and only if

$$\sum_{i=1}^m[x_i]=\sum_{i=1}^m[y_i]\text{ in }\operatorname{CH}_0(T)/F^{md+1}\operatorname{CH}_0(T).$$

The conjecture has been verified for several classes, including moduli spaces of stable objects on $K3$ surfaces, generalised Kummer varieties, Fano varieties of lines on cubic fourfolds, and O'Grady's six-dimensional hyperkähler varieties, for suitable candidate filtrations.