Boundary scaling limit conjecture for frozen-boundary diffusion

For frozen-boundary diffusion with parameter $\alpha$, let $\mu_t$ be its mass distribution and define the boundary by

$$\beta_t:=\sup\left\{x\in\mathbb Z:\mu_t([x,\infty))\geq\frac{\alpha}{2}\right\}.$$

Boundary scaling limit conjecture. For every $\alpha\in(0,1)$, there exists $\ell_{\alpha}>0$ such that

$$\lim_{t\to\infty}\frac{\beta_t}{\sqrt{t}}=\ell_{\alpha}.$$

The preceding bounds show that $\beta_t$ has order $\sqrt{t}$; the conjecture asks whether its normalized location converges. The source gives no resolution of this question.