Bressan's Fire Conjecture for isotropic fire blocking

Let the fire propagate with unit speed in all directions from $R_0=B_1(0)$, and let the barrier be constructed at constant speed $\sigma>0$, so that $F=\overline{B_1(0)}$ and $\psi=1/\sigma$. An admissible strategy is blocking when its burned region $R^\zeta_\infty=\bigcup_{t\geq 0}R^\zeta(t)$ is bounded. Bressan's Fire Conjecture. If $\sigma\leq 2$, no admissible blocking strategy exists. This conjecture identifies the expected sharp threshold between possible and impossible isotropic fire confinement; the result is known for $\sigma>2$ (existence) and for $\sigma\leq 1$ (nonexistence), leaving the intermediate range open.