The finite blocking number conjecture for compact Riemannian manifolds

Let $(M,g)$ be a compact Riemannian manifold. Its blocking number is the supremum, over pairs of points in $M$, of the minimal number of points needed to meet every geodesic segment joining the pair. A manifold has finite blocking number when this supremum is finite.

Finite blocking number conjecture. A compact Riemannian manifold has finite blocking number if and only if it is flat.

Compact flat manifolds are known to have finite blocking numbers, so the conjecture asks for the converse. The source notes that K. Burns and E. Gutkin obtained a partial solution; the full statement is not resolved here.