Conjectured traveling-wave limit for branching reflected Brownian motion

Let $u^R(t,x,y):=\mathbb{P}_y(M^R_t\leq x)$, where $\mathbb{P}_y$ is the law of branching reflected Brownian motion starting at $y\geq 0$. Let $q^R_{\delta}(y,t)$ be defined as in the source starting at $y$. The traveling-wave limit conjecture. For every $y\geq 0$, one has

$$u^R\bigl(t,x+q^R_{\delta}(y,t),y\bigr)\longrightarrow w^R(x),$$

where $w^R:\mathbb{R}^{+}\to[0,1]$ does not depend on $y$ and is the solution to a certain ODE to be specified. The conjecture concerns the long-time profile of the distribution of the frontier after centering by its median; the limiting ODE and the precise mode of convergence are not specified here.