Ballistic aggregation's stochastic differential approximation conjecture

Let $\mathcal{X}_n$ be the horizontal coordinate of the ballistic aggregation process, let $\theta$ be its parameter, and let $W$ be Brownian motion. Define

$$\mu(\theta)=\frac{\cos\theta}{1-\cos\theta},$$ $$\sigma(\theta)=\frac{\sqrt{\left(\theta^2+\theta\sin(\theta)\cos(\theta)\right)/2}}{1-\cos\theta}.$$

Stochastic differential approximation conjecture. The process $\mathcal{X}_n$ behaves like a continuous-time process $X_t$ satisfying

$$dX_t=\frac{\sigma(\theta)}{t}\,dW_t+\frac{\mu(\theta)X_t}{t}\,dt.$$

This conjecture formalizes the self-organized critical scaling suggested by the asymptotic drift and variance calculations. The supplied text does not state whether this approximation has been proved or disproved.