Conjecture on volume growth of the neighbourhood of the invasion percolation cluster

Let $T(\clock)$ be the branching process tree, and let

$$\mathcal{T}_k(\clock)=\{u\in T(\clock)\mid \text{height of }u<k\}$$

be the $k$-neighbourhood of the invasion percolation cluster. Under the conditions of the volume-growth theorem, let $x_k^{(\alpha)}$ denote the regime-dependent scaling of $M_k$, and let $K^{(\alpha)}$ be a non-degenerate limiting random process. Volume-growth conjecture. There exists such a sequence $x_k^{(\alpha)}$ and process $(K^{(\alpha)}(t))_{t>0}$ such that

$$\left(x_k^{(\alpha)}\left|\mathcal{T}_{\lceil kt\rceil}(\clock)\right|\right)_{t>0}\xrightarrow{\mathcal{D}}\left(K^{(\alpha)}(t)\right)_{t>0}$$

as $k\to\infty$, with $x_k^{(\alpha)}$ given by the scaling of $M_k$ in the main theorem. The conjecture asserts that the total volume of the $k$-neighbourhood has the same scaling behaviour as $M_k$; establishing the limiting processes and the required convergence remains open.