Vanishing averaged scalar-curvature conjecture for complete Kähler manifolds

Let $(M^n,p)$ be a complete noncompact Kähler manifold with nonnegative Ricci curvature. Write

$$\Xint-_{B(p,r)}S$$

for the average of the scalar curvature $S$ over the metric ball $B(p,r)$.

Averaged scalar-curvature conjecture. If

$$\lim_{r\to\infty}\inf r^2\Xint-_{B(p,r)}S=0,$$

then $M$ is Ricci flat. It is also asked whether the limit

$$\lim_{r\to\infty}r^2\Xint-_{B(p,r)}S$$

exists, possibly with value infinity.

The associated rigidity statement would extend the paper's Ricci-flatness criteria from stronger geometric assumptions to an averaged scalar-curvature condition. The existence of the unrestricted limit remains an additional open question.