Dynamic Blocking Problems of Fire Propagation

One of my main research interests is the study of dynamic blocking problems of fire propagation, which ultimately relate to the Bressan Fire Conjecture. Imagine a fire spreading uniformly in all directions from a closed ball in the plane. At the same time, firefighters attempt to contain the fire by constructing barriers. A central question is: What is the minimum speed the firefighters must have in order to successfully block the spread of the fire?

Recently, we proved that if a firefighter constructs a spiral-like barrier, the fire can be contained only if the firefighter's speed exceeds approximately 2.6144.. This raises a natural question: What happens when more general types of barriers are allowed?