The $N^{0.634}$ fluctuation conjecture for the height of $\mathbb{C}_0(t)$

Let $\mathbb{C}_0(t)$ denote the relevant boundary cluster at time $t$, and let $N$ be the scaling parameter used for the Poisson percolation model. Fluctuation conjecture. Fluctuations in the height of $\mathbb{C}_0(t)$ are of order $N^{0.634}$. This is a physicist-style prediction based on critical-exponent heuristics; the paper states that it is not proved mathematically because very little is known rigorously about critical exponents for oriented percolation.