Near-field scaling conjecture for the diffusion-off boundary

Let $v(x)$ be an antisymmetric velocity field with $v(x)>0$ for $x>0$, and suppose

$$\lim_{x\downarrow 0}\frac{\log v(x)}{\log x}=\beta>0.$$

Let $b(x)$ denote the diffusion-off boundary. Near-field scaling conjecture. Then

$$\lim_{x\downarrow 0}\frac{\log(-b(x))}{\log x}=\frac{1}{2}\left(\beta-1+\sqrt{\beta^2+2\beta+9}\right).$$

This predicts the near-origin power-law exponent of the boundary separating diffusion-on and diffusion-off regions. The paper derives the exponent heuristically by balancing two failure probabilities and compares it with numerical solutions, but does not establish the asserted limit analytically.