Linear growth conjecture for maximal facet dimensions in Eulerian magnitude homology

Let $G$ be a graph, let $\operatorname{diam}(G)$ denote its diameter, and let $\ell$ be the path-length parameter. Write $f_t$ for the facet whose dimension is under consideration. Here $\dim f_t\sim\ell$ means that the dimension of $f_t$ is asymptotic to $\ell$.

Linear growth conjecture. There exists a linear function $\varphi$ such that, whenever

$$\ell\leq\varphi(\operatorname{diam}(G)),$$

one has

$$\dim f_t\sim\ell.$$

The conjecture proposes that the relevant facet dimension grows linearly with the length parameter when that parameter is bounded linearly in the graph diameter. It is motivated by computations for path graphs and is stated in the context of estimating torsion in Eulerian magnitude homology of random graphs; no resolution is supplied here.