Metric conjecture for affine preferential attachment seed graphs

Fix a parameter $\delta>-1$. Let $S$ be a finite tree with $n_0$ vertices, and let $({T}^{(S),\delta}_n)_{n\geq n_0}$ be the affine preferential attachment tree sequence started from $S$, where each new vertex attaches to $u$ with probability proportional to $\deg(u)+\delta$. For two trees $S_1$ and $S_2$, define

$$d_\delta(S_1,S_2)=\lim_{n\to\infty}\mathrm{d_{TV}}({T}^{(S_1),\delta}_n,{T}^{(S_2),\delta}_n).$$

Affine preferential attachment metric conjecture. The function $d_\delta$ is a metric on trees with at least $3$ vertices.

This extends the seed-influence question from the linear model to affine reinforcement. The source suggests using the same embedding observables as in the linear case, but does not pursue a proof.