Beauville's filtration-compatibility conjecture for the Grothendieck group

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field of characteristic zero, and let $K_0(A)\otimes\mathbf Q$ carry its usual gamma filtration $\operatorname{Fil}^q_\gamma$ and the Pontryagin filtration $\operatorname{Fil}^q_\pi$ arising from the Pontryagin lambda-ring structure. Beauville's filtration-compatibility conjecture.

$$\operatorname{Fil}^q_\pi\subseteq\operatorname{Fil}^q_\gamma\qquad\text{for all }0\leq q\leq g.$$

The paper presents this as an equivalent formulation of Beauville's conjecture on algebraic cycles. It notes that the inclusion is known for $q=0,1,g-1,g$, while the general case remains open.