The cubic-order conjecture for bridge-burning capture time

Let $G$ be an $n$-vertex graph. Write $c_b(G)$ for its bridge-burning cop number and $\mathrm{capt}_b(G)$ for its bridge-burning capture time. Cubic-order conjecture. There exists an $n$-vertex graph $G$ with $c_b(G)=1$ and

$$\mathrm{capt}_b(G)=\Omega(n^3).$$

The preceding construction establishes only a quadratic lower bound, while the paper's upper bound differs by an order of magnitude; the conjecture asserts that the upper bound has the correct order of growth.