The Maki-Thompson model describes rumor spreading among individuals arranged on an infinite graph, where spreaders contact neighbors and either convert them or retire from spreading. On trees and high-dimensional lattices, survival versus extinction behavior is relatively tractable, but on the two-dimensional integer lattice the geometry creates subtle correlations between spreaders that make the long-run fate of the rumor extremely difficult to pin down. The core open question is whether there exists a single critical threshold for the transmission parameter below which the rumor dies out almost surely and above which it survives with positive probability, and whether this transition is genuinely sharp in the sense of having no intermediate phase.