Scaling-limit conjecture for the breadth-first walk on critical Erdős–Rényi graphs

Let $\theta_n$ satisfy the scaling condition in the source, and let $X_n=(X_n(k);k=0,1,\dotsm)$ be the breadth-first walk on the random graph $\mathscr{G}_n^\theta$ constructed for the $\mathscr{G}_n$ model. The process is viewed in the Skorohod space $\mathbb{D}(\mathbb{R}_+,\mathbb{R})$. Breadth-first-walk scaling-limit conjecture. The following convergence holds:

$$\left(\frac{1}{n^{1/3}\theta_n^2}X_n\left(\left\lfloor n^{2/3}\theta_n t\right\rfloor\right);t\ge 0\right)\Longrightarrow\left(\lambda t-\frac{1}{2}t^2;t\ge 0\right).$$

This conjecture asks whether the breadth-first walk associated with the critical Erdős–Rényi graph has the same parabolic-drift scaling limit suggested by the continuous Lamperti transform. The paper does not establish this convergence for the breadth-first walk, so the claim remains open.