Scaling-limit conjecture for the unlabelled $(\alpha,\gamma)$-chain

Let $0 < \gamma \leq \alpha < 1$. Consider the unlabelled $(\alpha,\gamma)$-chain on rooted multifurcating trees with $n$ leaves, obtained by selecting and deleting a uniform leaf and inserting a new leaf according to the unlabelled $(\alpha,\gamma)$-growth rule. Represent its edges at scale $n^\gamma$, and view the resulting weighted trees in the Gromov-Hausdorff-Prokhorov space of weighted $\mathbb{R}$-trees. The unlabelled $(\alpha,\gamma)$-chain scaling-limit conjecture. The unlabelled $(\alpha,\gamma)$-chain with $n^2$ steps per time unit and edges scaled by $n^\gamma$ has a scaling limit that is a path-continuous Markov process in the Gromov-Hausdorff-Prokhorov space of weighted $\mathbb{R}$-trees, whose stationary distribution is that of $\mathcal{T}^{\alpha,\gamma}$. The conjecture extends the finite-dimensional Wright-Fisher diffusion limit for decorated projections and the stationary Gromov-Hausdorff-Prokhorov convergence to a dynamical continuum-tree limit; neither the asserted process-level convergence nor the required path continuity is established in the supplied text.