The general-graph extension of the estimation-error lower-bound lemma

Let $G$ satisfy the paper's general-graph regularity assumption, and let $\\{V_n\\}_{n \ge 1}$ satisfy the candidate-set assumption. For each initial source $v$, write $\mathbb{P}_v$ for probability under source $v$, let $v^*$ be the true source, let $\widehat{v}_B(t)$ be the estimator at time $t$, let $\pi(t)$ be the observation history up to time $t$, let $d$ be graph distance, and let $F_v$ denote the relevant graph-dependent time-scaling function. General-graph estimation-error conjecture. There are constants $a_1”=a_1”(G,Q_0,Q_1)$ and $b_1”=b_1”(G)$ such that

$$\lim_{n\to\infty}\max_{v\in V_n}\mathbb{P}_v\left(\min_{0\le t\le F_v(a_1”\log n)}\mathbb{E}_{\pi(t)}[d(v^*,\widehat{v}_B(t))]\le b_1”\cdot\mathbb{E}_{\pi(0)}[d(v^*,\widehat{v}_B(0))]\right)=0.$$

The claim is presented as an expected extension of the paper's lower-bound lemma from regular trees and lattices to suitable general graphs; the displayed text leaves the referenced graph and candidate-set assumptions unspecified, so its precise scope and eventual status require verification.