Weak-limit conjecture for frozen-boundary diffusion

Fix $\alpha\in(0,1)$. Let $\mu_t$ be the mass distribution of frozen-boundary diffusion and define the rescaled Borel probability measure $\tilde\mu_t$ on $\mathbb R$ by

$$\tilde\mu_t(A)=\mu_t(\{y\sqrt{t}:y\in A\})$$

for every Borel set $A\subseteq\mathbb R$. Here $\tilde\mu_t\mathrel{\stackrel{\mathrm{weak}}{\implies}}\mu$ denotes weak convergence in the space of finite measures on $\mathbb R$.

Weak-limit conjecture. There exists a probability measure $\mu_{\infty}(\alpha)$ on $\mathbb R$ such that, as $t\to\infty$,

$$\tilde\mu_t\mathrel{\stackrel{\mathrm{weak}}{\implies}}\mu_{\infty}(\alpha).$$

The conjecture is the affirmative answer to the question of whether the rescaled mass distributions have a weak limit. The source does not provide a limiting distribution or evidence that the conjecture has been resolved.