The two 1-visibility cops cleaning conjecture

Let $G$ be a connected graph. Two 1-visibility cops are cops whose visibility parameter is $1$, and a graph is cleaned when all its vertices are visited according to the rules of the limited-visibility cops and robbers game.

Two 1-visibility cops cleaning conjecture. Two 1-visibility cops can clean at least $10$ vertices of $G$, or they can clean the whole graph.

This conjecture proposes that $10$ is the maximum guaranteed number of cleaned vertices for two 1-visibility cops unless the entire connected graph can be cleaned. The paper reports computational evidence for connected graphs up to $10$ vertices and notes the Heawood graph as an example that cannot be completely cleaned by two such cops.